3.24.60 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx\) [2360]

Optimal. Leaf size=238 \[ -\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}+\frac {23466191827 \sqrt {1-2 x} \sqrt {3+5 x}}{4182119424 (2+3 x)}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {1104970911 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17210368 \sqrt {7}} \]

[Out]

-1/21*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7-1104970911/120472576*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2
))*7^(1/2)-12421/52920*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5+181/756*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^6-128
9227/8890560*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+6249601/53343360*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+2240
18941/298722816*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+23466191827/4182119424*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*
x)

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Rubi [A]
time = 0.07, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 154, 156, 12, 95, 210} \begin {gather*} -\frac {1104970911 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{17210368 \sqrt {7}}+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{756 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac {12421 \sqrt {1-2 x} (5 x+3)^{3/2}}{52920 (3 x+2)^5}+\frac {23466191827 \sqrt {1-2 x} \sqrt {5 x+3}}{4182119424 (3 x+2)}+\frac {224018941 \sqrt {1-2 x} \sqrt {5 x+3}}{298722816 (3 x+2)^2}+\frac {6249601 \sqrt {1-2 x} \sqrt {5 x+3}}{53343360 (3 x+2)^3}-\frac {1289227 \sqrt {1-2 x} \sqrt {5 x+3}}{8890560 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

(-1289227*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8890560*(2 + 3*x)^4) + (6249601*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(53343360
*(2 + 3*x)^3) + (224018941*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(298722816*(2 + 3*x)^2) + (23466191827*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(4182119424*(2 + 3*x)) - (12421*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(52920*(2 + 3*x)^5) - ((1 - 2*x)^
(3/2)*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^7) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(756*(2 + 3*x)^6) - (1104970911*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(17210368*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 99

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

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 210

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

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {1}{378} \int \frac {(3+5 x)^{3/2} \left (-\frac {6461}{4}+2235 x\right )}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {\int \frac {\sqrt {3+5 x} \left (-\frac {693747}{8}+\frac {223305 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{39690}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {\int \frac {-\frac {31720047}{16}+\frac {4510185 x}{4}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{3333960}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {\int \frac {-\frac {4340886375}{32}+\frac {656208105 x}{4}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{70013160}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {\int \frac {-\frac {507690196545}{64}+\frac {117609944025 x}{16}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{980184240}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}+\frac {23466191827 \sqrt {1-2 x} \sqrt {3+5 x}}{4182119424 (2+3 x)}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {\int -\frac {28193332794165}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6861289680}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}+\frac {23466191827 \sqrt {1-2 x} \sqrt {3+5 x}}{4182119424 (2+3 x)}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}+\frac {1104970911 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{34420736}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}+\frac {23466191827 \sqrt {1-2 x} \sqrt {3+5 x}}{4182119424 (2+3 x)}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}+\frac {1104970911 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{17210368}\\ &=-\frac {1289227 \sqrt {1-2 x} \sqrt {3+5 x}}{8890560 (2+3 x)^4}+\frac {6249601 \sqrt {1-2 x} \sqrt {3+5 x}}{53343360 (2+3 x)^3}+\frac {224018941 \sqrt {1-2 x} \sqrt {3+5 x}}{298722816 (2+3 x)^2}+\frac {23466191827 \sqrt {1-2 x} \sqrt {3+5 x}}{4182119424 (2+3 x)}-\frac {12421 \sqrt {1-2 x} (3+5 x)^{3/2}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{756 (2+3 x)^6}-\frac {1104970911 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17210368 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 96, normalized size = 0.40 \begin {gather*} \frac {161051 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (33120084096+294736348384 x+1092179419888 x^2+2158260396608 x^3+2399706883464 x^4+1423652835490 x^5+351992877405 x^6\right )}{161051 (2+3 x)^7}-34305 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{602362880} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

(161051*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(33120084096 + 294736348384*x + 1092179419888*x^2 + 2158260396608*x^3
+ 2399706883464*x^4 + 1423652835490*x^5 + 351992877405*x^6))/(161051*(2 + 3*x)^7) - 34305*Sqrt[7]*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/602362880

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(187)=374\).
time = 0.13, size = 394, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (351992877405 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{86051840 \left (2+3 x \right )^{7} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1104970911 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{240945152 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(144\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (12082856911785 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}[phone] \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{1204725760 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\)	\(394\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^8,x,method=_RETURNVERBOSE)

[Out]

1/1204725760*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(12082856911785*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^7+56386665588330*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^6+112773331176660*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+4927900283670*(-10*x^2-x+3)^(1/2)*x^6+125303701307400
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+19931139696860*x^5*(-10*x^2-x+3)^(1/2)+8353580
0871600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+33595896368496*x^4*(-10*x^2-x+3)^(1/2)+
33414320348640*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+30215645552512*x^3*(-10*x^2-x+3)
^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+15290511878432*x^2*(-10*x^2-
x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+
3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^7

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Maxima [A]
time = 0.53, size = 324, normalized size = 1.36 \begin {gather*} \frac {207419465}{90354432} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{49 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {157 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{4116 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {6289 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{41160 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {75471 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2792427 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2151296 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {124451679 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1689418335}{60236288} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1104970911}{240945152} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1488514533}{120472576} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {492397961 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{361417728 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="maxima")

[Out]

207419465/90354432*(-10*x^2 - x + 3)^(3/2) - 1/49*(-10*x^2 - x + 3)^(5/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 +
22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 157/4116*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860
*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 6289/41160*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 +
720*x^2 + 240*x + 32) + 75471/153664*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 279242
7/2151296*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 124451679/60236288*(-10*x^2 - x + 3)^(5/2)/(9
*x^2 + 12*x + 4) + 1689418335/60236288*sqrt(-10*x^2 - x + 3)*x + 1104970911/240945152*sqrt(7)*arcsin(37/11*x/a
bs(3*x + 2) + 20/11/abs(3*x + 2)) - 1488514533/120472576*sqrt(-10*x^2 - x + 3) + 492397961/361417728*(-10*x^2
- x + 3)^(3/2)/(3*x + 2)

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Fricas [A]
time = 0.94, size = 161, normalized size = 0.68 \begin {gather*} -\frac {5524854555 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (351992877405 \, x^{6} + 1423652835490 \, x^{5} + 2399706883464 \, x^{4} + 2158260396608 \, x^{3} + 1092179419888 \, x^{2} + 294736348384 \, x + 33120084096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1204725760 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/1204725760*(5524854555*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*
x + 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(351992877405*x^6
+ 1423652835490*x^5 + 2399706883464*x^4 + 2158260396608*x^3 + 1092179419888*x^2 + 294736348384*x + 3312008409
6)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x
+ 128)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
time = 2.71, size = 542, normalized size = 2.28 \begin {gather*} \frac {1104970911}{2409451520} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (6861 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{13} + 12807200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 10148425280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 3461100339200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 785566018048000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 78720223232000000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {3306249375744000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {13224997502976000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{8605184 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="giac")

[Out]

1104970911/2409451520*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 161051/8605184*sqrt(10)*(6861*((sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^13 + 128072
00*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))
^11 + 10148425280*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x +
5) - sqrt(22)))^9 - 3461100339200*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 785566018048000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 78720223232000000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/
sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 3306249375744000000*(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) + 13224997502976000000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/
(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^8,x)

[Out]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(5/2))/(3*x + 2)^8, x)

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