3.23.13 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^3} \, dx\)

Optimal. Leaf size=181 \[ -\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac {7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac {7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac {7 (167495-349806 x) \sqrt {3 x^2+5 x+2}}{36864}-\frac {12443893 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024} \]

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Rubi [A]  time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {812, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(x+21) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^2}+\frac {7 (121 x+584) \left (3 x^2+5 x+2\right )^{5/2}}{240 (2 x+3)}+\frac {7 (805-17394 x) \left (3 x^2+5 x+2\right )^{3/2}}{4608}+\frac {7 (167495-349806 x) \sqrt {3 x^2+5 x+2}}{36864}-\frac {12443893 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(167495 - 349806*x)*Sqrt[2 + 5*x + 3*x^2])/36864 + (7*(805 - 17394*x)*(2 + 5*x + 3*x^2)^(3/2))/4608 + (7*(5
84 + 121*x)*(2 + 5*x + 3*x^2)^(5/2))/(240*(3 + 2*x)) - ((21 + x)*(2 + 5*x + 3*x^2)^(7/2))/(12*(3 + 2*x)^2) - (
12443893*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(73728*Sqrt[3]) + (44625*Sqrt[5]*ArcTanh[(7 + 8
*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

Rule 206

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

Rule 621

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

Rule 724

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 812

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

Rule 814

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^3} \, dx &=-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {7}{96} \int \frac {(-404-484 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx\\ &=\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac {7}{768} \int \frac {(-19488-23192 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {7 \int \frac {(2361672+2798448 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{73728}\\ &=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}+\frac {7 \int \frac {-145828656-170659104 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{3538944}\\ &=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{73728}+\frac {223125 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1024}\\ &=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{36864}-\frac {223125}{512} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {7 (167495-349806 x) \sqrt {2+5 x+3 x^2}}{36864}+\frac {7 (805-17394 x) \left (2+5 x+3 x^2\right )^{3/2}}{4608}+\frac {7 (584+121 x) \left (2+5 x+3 x^2\right )^{5/2}}{240 (3+2 x)}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^2}-\frac {12443893 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{73728 \sqrt {3}}+\frac {44625 \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 130, normalized size = 0.72 \begin {gather*} \frac {-48195000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-62219465 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {6 \sqrt {3 x^2+5 x+2} \left (414720 x^7-926208 x^6-6830784 x^5-15112992 x^4-12848072 x^3-19284852 x^2-89867034 x-91912653\right )}{(2 x+3)^2}}{1105920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(-91912653 - 89867034*x - 19284852*x^2 - 12848072*x^3 - 15112992*x^4 - 6830784*x^5
- 926208*x^6 + 414720*x^7))/(3 + 2*x)^2 - 48195000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)] - 62219465*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/1105920

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IntegrateAlgebraic [A]  time = 0.95, size = 131, normalized size = 0.72 \begin {gather*} -\frac {12443893 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{36864 \sqrt {3}}+\frac {44625}{512} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-414720 x^7+926208 x^6+6830784 x^5+15112992 x^4+12848072 x^3+19284852 x^2+89867034 x+91912653\right )}{184320 (2 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(91912653 + 89867034*x + 19284852*x^2 + 12848072*x^3 + 15112992*x^4 + 6830784*x^5 + 926
208*x^6 - 414720*x^7))/(184320*(3 + 2*x)^2) - (12443893*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(368
64*Sqrt[3]) + (44625*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/512

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fricas [A]  time = 0.44, size = 173, normalized size = 0.96 \begin {gather*} \frac {62219465 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 48195000 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\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 ) - 12 \, {\left (414720 \, x^{7} - 926208 \, x^{6} - 6830784 \, x^{5} - 15112992 \, x^{4} - 12848072 \, x^{3} - 19284852 \, x^{2} - 89867034 \, x - 91912653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{2211840 \, {\left (4 \, x^{2} + 12 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/2211840*(62219465*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x
+ 49) + 48195000*sqrt(5)*(4*x^2 + 12*x + 9)*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)) - 12*(414720*x^7 - 926208*x^6 - 6830784*x^5 - 15112992*x^4 - 12848072*x^3 - 19284852
*x^2 - 89867034*x - 91912653)*sqrt(3*x^2 + 5*x + 2))/(4*x^2 + 12*x + 9)

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giac [A]  time = 0.38, size = 279, normalized size = 1.54 \begin {gather*} -\frac {1}{184320} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 157\right )} x - 725\right )} x - 67409\right )} x + 1173065\right )} x - 8219517\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {44625}{1024} \, \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 {12443893}{221184} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {25 \, {\left (5878 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 22241 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 75807 \, \sqrt {3} x + 27061 \, \sqrt {3} - 75807 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{512 \, {\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 )}^{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)^3,x, algorithm="giac")

[Out]

-1/184320*(2*(12*(18*(8*(30*x - 157)*x - 725)*x - 67409)*x + 1173065)*x - 8219517)*sqrt(3*x^2 + 5*x + 2) + 446
25/1024*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*s
qrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 12443893/221184*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5)) + 25/512*(5878*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 22241*sqrt(3)*(sqrt(3)*x - sqrt
(3*x^2 + 5*x + 2))^2 + 75807*sqrt(3)*x + 27061*sqrt(3) - 75807*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)^2

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maple [A]  time = 0.06, size = 253, normalized size = 1.40 \begin {gather*} -\frac {44625 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1024}-\frac {12443893 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{221184}+\frac {27 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10 \left (x +\frac {3}{2}\right )}+\frac {51 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{8}-\frac {1127 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{480}-\frac {20293 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{4608}-\frac {408107 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{36864}+\frac {357 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{32}+\frac {2975 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{128}+\frac {44625 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1024}-\frac {27 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{20}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/10/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+51/8*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-1127/480*(6*x+5)*(-4*x+3*(x+3/2
)^2-19/4)^(5/2)-20293/4608*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-408107/36864*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^
(1/2)-12443893/221184*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))+357/32*(-4*x+3*(x+3/2)^2
-19/4)^(5/2)+2975/128*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+44625/1024*(-16*x+12*(x+3/2)^2-19)^(1/2)-44625/1024*5^(1/2
)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-27/20*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-13
/40/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)

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maxima [A]  time = 1.14, size = 218, normalized size = 1.20 \begin {gather*} \frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1127}{80} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {7}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{4 \, {\left (2 \, x + 3\right )}} - \frac {20293}{768} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5635}{4608} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {408107}{6144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {12443893}{221184} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {44625}{1024} \, \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 {1172465}{36864} \, \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)^3,x, algorithm="maxima")

[Out]

39/40*(3*x^2 + 5*x + 2)^(7/2) - 13/10*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 1127/80*(3*x^2 + 5*x + 2)^(
5/2)*x - 7/12*(3*x^2 + 5*x + 2)^(5/2) + 27/4*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 20293/768*(3*x^2 + 5*x + 2)^(
3/2)*x + 5635/4608*(3*x^2 + 5*x + 2)^(3/2) - 408107/6144*sqrt(3*x^2 + 5*x + 2)*x - 12443893/221184*sqrt(3)*log
(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 44625/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3
) + 5/2/abs(2*x + 3) - 2) + 1172465/36864*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^3} \,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)^3,x)

[Out]

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

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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}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, 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)**3,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-292*x*sqrt(3*x**2 + 5*x +
2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 2
7), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-1090*x**4*s
qrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3 +
36*x**2 + 54*x + 27), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x)

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