3.25.64 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^{13}} \, dx\) [2464]

Optimal. Leaf size=259 \[ -\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000000 \sqrt {5}} \]

[Out]

58373/512000000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-58373/32000000*(7+8*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6+250
17/800000*(7+8*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8-13/60*(3*x^2+5*x+2)^(9/2)/(3+2*x)^12-12/55*(3*x^2+5*x+2)^(9/2)
/(3+2*x)^11-2067/11000*(3*x^2+5*x+2)^(9/2)/(3+2*x)^10-6379/41250*(3*x^2+5*x+2)^(9/2)/(3+2*x)^9+175119/20480000
0000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-175119/20480000000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/
(3+2*x)^2

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Rubi [A]
time = 0.12, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {848, 820, 734, 738, 212} \begin {gather*} -\frac {6379 \left (3 x^2+5 x+2\right )^{9/2}}{41250 (2 x+3)^9}-\frac {2067 \left (3 x^2+5 x+2\right )^{9/2}}{11000 (2 x+3)^{10}}-\frac {12 \left (3 x^2+5 x+2\right )^{9/2}}{55 (2 x+3)^{11}}-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}+\frac {25017 (8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{800000 (2 x+3)^8}-\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{32000000 (2 x+3)^6}+\frac {58373 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{512000000 (2 x+3)^4}-\frac {175119 (8 x+7) \sqrt {3 x^2+5 x+2}}{20480000000 (2 x+3)^2}+\frac {175119 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40960000000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]

[Out]

(-175119*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20480000000*(3 + 2*x)^2) + (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(512000000*(3 + 2*x)^4) - (58373*(7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(32000000*(3 + 2*x)^6) + (25017*(7 + 8*x
)*(2 + 5*x + 3*x^2)^(7/2))/(800000*(3 + 2*x)^8) - (13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) - (12*(2 + 5*
x + 3*x^2)^(9/2))/(55*(3 + 2*x)^11) - (2067*(2 + 5*x + 3*x^2)^(9/2))/(11000*(3 + 2*x)^10) - (6379*(2 + 5*x + 3
*x^2)^(9/2))/(41250*(3 + 2*x)^9) + (175119*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(40960000000*
Sqrt[5])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {1}{60} \int \frac {\left (-\frac {369}{2}+117 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{12}} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}+\frac {\int \frac {\left (\frac {18045}{2}-4320 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{11}} \, dx}{3300}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {\int \frac {\left (-\frac {869175}{2}+93015 x\right ) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{10}} \, dx}{165000}\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {25017 \int \frac {\left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^9} \, dx}{10000}\\ &=\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \int \frac {\left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx}{1600000}\\ &=-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {58373 \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{12800000}\\ &=\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{1024000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{40960000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}-\frac {175119 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{20480000000}\\ &=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.90, size = 113, normalized size = 0.44 \begin {gather*} \frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (2531527640959+25843081681156 x+111795175925940 x^2+271870111600160 x^3+412855931529440 x^4+410468875350912 x^5+273282692080768 x^6+123629135656960 x^7+38544695427840 x^8+8182662620160 x^9+1044584776704 x^{10}[phone] x^{11}\right )}{(3+2 x)^{12}}+5778927 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{3379200000000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^13,x]

[Out]

((5*Sqrt[2 + 5*x + 3*x^2]*(2531527640959 + 25843081681156*x + 111795175925940*x^2 + 271870111600160*x^3 + 4128
55931529440*x^4 + 410468875350912*x^5 + 273282692080768*x^6 + 123629135656960*x^7 + 38544695427840*x^8 + 81826
62620160*x^9 + 1044584776704*x^10 + 60734693376*x^11))/(3 + 2*x)^12 + 5778927*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (
3*x^2)/5]/(1 + x)])/3379200000000

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Maple [A]
time = 0.18, size = 432, normalized size = 1.67

method	result	size
risch	\(\frac {182204080128 x^{13}[phone] x^{12}[phone] x^{11}[phone] x^{10}[phone] x^{9}+1515083145382784 x^{8}+2845078357770496 x^{7}+3837477555504416 x^{6}+3700827743149504 x^{5}+2520447948837500 x^{4}+1180245347873488 x^{3}[phone] x^{2}[phone] x [phone]}{675840000000 \left (3+2 x \right )^{12} \sqrt {3 x^{2}+5 x +2}}-\frac {175119 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{204800000000}\)	\(118\)
trager	\(\frac {\left (60734693376 x^{11}[phone] x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}+5 x +2}}{675840000000 \left (3+2 x \right )^{12}}-\frac {175119 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {8 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +7 \RootOf \left (\textit {\_Z}^{2}-5\right )-10 \sqrt {3 x^{2}+5 x +2}}{3+2 x}\right )}{204800000000}\)	\(128\)
default	\(\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{200000000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{245760 \left (x +\frac {3}{2}\right )^{12}}+\frac {175119 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{204800000000}-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{28160 \left (x +\frac {3}{2}\right )^{11}}-\frac {2067 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{11264000 \left (x +\frac {3}{2}\right )^{10}}-\frac {6379 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{21120000 \left (x +\frac {3}{2}\right )^{9}}-\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{51200000 \left (x +\frac {3}{2}\right )^{8}}-\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{32000000 \left (x +\frac {3}{2}\right )^{7}}-\frac {158441 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{128000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {775527 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{400000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {48057657 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{16000000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {46022941 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1395223107 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{200000000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {101744139 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200000000000}+\frac {1692817 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{32000000000}+\frac {261602769 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{50000000000}-\frac {261602769 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{25000000000 \left (x +\frac {3}{2}\right )}-\frac {175119 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{25600000000}-\frac {175119 \sqrt {5}\, \arctanh \left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{204800000000}+\frac {58373 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{128000000000}+\frac {175119 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{800000000000}\)	\(432\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x,method=_RETURNVERBOSE)

[Out]

25017/200000000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)-13/245760/(x+3/2)^12*(3*(x+3/2)^2-4*x-19/4)^(9/2)+175119/20480
0000000*(12*(x+3/2)^2-16*x-19)^(1/2)-3/28160/(x+3/2)^11*(3*(x+3/2)^2-4*x-19/4)^(9/2)-2067/11264000/(x+3/2)^10*
(3*(x+3/2)^2-4*x-19/4)^(9/2)-6379/21120000/(x+3/2)^9*(3*(x+3/2)^2-4*x-19/4)^(9/2)-25017/51200000/(x+3/2)^8*(3*
(x+3/2)^2-4*x-19/4)^(9/2)-25017/32000000/(x+3/2)^7*(3*(x+3/2)^2-4*x-19/4)^(9/2)-158441/128000000/(x+3/2)^6*(3*
(x+3/2)^2-4*x-19/4)^(9/2)-775527/400000000/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-48057657/16000000000/(x+3/2)
^4*(3*(x+3/2)^2-4*x-19/4)^(9/2)-46022941/10000000000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1395223107/2000000
00000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-101744139/200000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+16928
17/32000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)[phone]/50000000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)
-261602769/25000000000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)-175119/25600000000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^
(1/2)-175119/204800000000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))+58373/128000000
000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+175119/800000000000*(3*(x+3/2)^2-4*x-19/4)^(5/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (217) = 434\).
time = 0.51, size = 726, normalized size = 2.80 \begin {gather*} \frac {4185669321}{200000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{60 \, {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} - \frac {12 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{55 \, {\left (2048 \, x^{11} + 33792 \, x^{10} + 253440 \, x^{9} + 1140480 \, x^{8} + 3421440 \, x^{7} + 7185024 \, x^{6} + 10777536 \, x^{5} + 11547360 \, x^{4} + 8660520 \, x^{3} + 4330260 \, x^{2} + 1299078 \, x + 177147\right )}} - \frac {2067 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{11000 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} - \frac {6379 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{41250 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {25017 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{200000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {25017 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {158441 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{2000000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {775527 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{12500000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {48057657 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {46022941 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1250000000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1395223107 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50000000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {305232417}{100000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {2034707661}{800000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {261602769 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{10000000000 \, {\left (2 \, x + 3\right )}} + \frac {5078451}{16000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {33914713}{128000000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {525357}{12800000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175119}{204800000000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {3327261}{102400000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="maxima")

[Out]

4185669321/200000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/60*(3*x^2 + 5*x + 2)^(9/2)/(4096*x^12 + 73728*x^11 + 6082
56*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^
3 + 15588936*x^2 + 4251528*x + 531441) - 12/55*(3*x^2 + 5*x + 2)^(9/2)/(2048*x^11 + 33792*x^10 + 253440*x^9 +
1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x
+ 177147) - 2067/11000*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6
+ 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 6379/41250*(3*x^2 + 5*x + 2)^(9/
2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x
+ 19683) - 25017/200000*(3*x^2 + 5*x + 2)^(9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108
864*x^3 + 81648*x^2 + 34992*x + 6561) - 25017/250000*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 +
15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 158441/2000000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5
+ 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 775527/12500000*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4
+ 720*x^3 + 1080*x^2 + 810*x + 243) - 48057657/1000000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81) - 46022941/1250000000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1395223107/500000000
00*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 305232417/100000000000*(3*x^2 + 5*x + 2)^(5/2)*x - 2034707661/
800000000000*(3*x^2 + 5*x + 2)^(5/2) - 261602769/10000000000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 5078451/16000
000000*(3*x^2 + 5*x + 2)^(3/2)*x + 33914713/128000000000*(3*x^2 + 5*x + 2)^(3/2) - 525357/12800000000*sqrt(3*x
^2 + 5*x + 2)*x - 175119/204800000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3
) - 2) - 3327261/102400000000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]
time = 2.68, size = 245, normalized size = 0.95 \begin {gather*} \frac {5778927 \, \sqrt {5} {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (60734693376 \, x^{11} + 1044584776704 \, x^{10} + 8182662620160 \, x^{9} + 38544695427840 \, x^{8} + 123629135656960 \, x^{7} + 273282692080768 \, x^{6} + 410468875350912 \, x^{5} + 412855931529440 \, x^{4} + 271870111600160 \, x^{3} + 111795175925940 \, x^{2} + 25843081681156 \, x + 2531527640959\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{13516800000000 \, {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="fricas")

[Out]

1/13516800000000*(5778927*sqrt(5)*(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 2463436
8*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441)*log((4*
sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 20*(60734693376*x^11 + 1
044584776704*x^10 + 8182662620160*x^9 + 38544695427840*x^8 + 123629135656960*x^7 + 273282692080768*x^6 + 41046
8875350912*x^5 + 412855931529440*x^4 + 271870111600160*x^3 + 111795175925940*x^2 + 25843081681156*x + 25315276
40959)*sqrt(3*x^2 + 5*x + 2))/(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^
7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**13,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**
9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 55269
864*x**2 + 13817466*x + 1594323), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 143
7696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5
+ 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integral(-870*x**2*sqrt(3*x**2
+ 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 16012
3392*x**7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1
594323), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*
x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 13
5104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(8192*x*
*13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 24018508
8*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integr
al(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**9
+ 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 552698
64*x**2 + 13817466*x + 1594323), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 143
7696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5
+ 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (217) = 434\).
time = 0.91, size = 716, normalized size = 2.76 \begin {gather*} \frac {175119}{204800000000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {11835242496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{23} + 408315866112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{22} + 20039038086144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{21} + 535243596890112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{20} + 13859706456921600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 31535346744025344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} - 789031961976842496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} - 7977976824329385984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 113078650509677476096 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 358779889050339715200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 2538162771649151164032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 4660243350382625915904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 20499122524155108829248 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 24347916060701730772704 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 70788415443572756925600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 56076083911431114398208 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 108598043564223524909928 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 56663550021725424101412 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 70668287639831997261828 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 22876037084903247115200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 16680770211437743348146 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 2864949797863813201587 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 930278306769206446269 \, \sqrt {3} x - 47729262032858665512 \, \sqrt {3} + 930278306769206446269 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{675840000000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^13,x, algorithm="giac")

[Out]

175119/204800000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqr
t(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/675840000000*(11835242496*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^23 + 408315866112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^22 + 20039038086144*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^21 + 535243596890112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 13859706456921600
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 31535346744025344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 - 7
89031961976842496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 7977976824329385984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^16 - 113078650509677476096*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 358779889050339715200*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 2538162771649151164032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 4660
243350382625915904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 20499122524155108829248*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^11 - 24347916060701730772704*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 70788415443572
756925600*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 56076083911431114398208*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*
x + 2))^8 - 108598043564223524909928*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 56663550021725424101412*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 70668287639831997261828*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 2287603
7084903247115200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 16680770211437743348146*(sqrt(3)*x - sqrt(3*x
^2 + 5*x + 2))^3 - 2864949797863813201587*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 93027830676920644626
9*sqrt(3)*x - 47729262032858665512*sqrt(3) + 930278306769206446269*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^12

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{13}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^13,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^13, x)

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