3.2627 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=171 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{511 (3 x+2)^4}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7591 \sqrt{1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac{261331 \sqrt{1-2 x} (3 x+2)^2}{2196150 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (78981180 x+190406711)}{117128000}+\frac{753543 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

[Out]

(7591*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(39930*(3 + 5*x)^(3/2)) - (511*(2 + 3*x)^4)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (261331*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(2196150*Sqrt[3
+ 5*x]) - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x))/117128000 + (753543*ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]])/(8000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0573238, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{511 (3 x+2)^4}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7591 \sqrt{1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac{261331 \sqrt{1-2 x} (3 x+2)^2}{2196150 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (78981180 x+190406711)}{117128000}+\frac{753543 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(7591*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(39930*(3 + 5*x)^(3/2)) - (511*(2 + 3*x)^4)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (261331*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(2196150*Sqrt[3
+ 5*x]) - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x))/117128000 + (753543*ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]])/(8000*Sqrt[10])

Rule 98

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

Rule 150

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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 54

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

Rule 216

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

Rubi steps

\begin{align*} \int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^4 \left (204+\frac{717 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-\frac{32415}{2}-\frac{115641 x}{4}\right ) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{7591 \sqrt{1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\left (-880026-\frac{11995011 x}{8}\right ) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{7591 \sqrt{1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{261331 \sqrt{1-2 x} (2+3 x)^2}{2196150 \sqrt{3+5 x}}-\frac{4 \int \frac{\left (-\frac{127241163}{8}-\frac{414651195 x}{16}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3294225}\\ &=\frac{7591 \sqrt{1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{261331 \sqrt{1-2 x} (2+3 x)^2}{2196150 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (190406711+78981180 x)}{117128000}+\frac{753543 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{16000}\\ &=\frac{7591 \sqrt{1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{261331 \sqrt{1-2 x} (2+3 x)^2}{2196150 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (190406711+78981180 x)}{117128000}+\frac{753543 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8000 \sqrt{5}}\\ &=\frac{7591 \sqrt{1-2 x} (2+3 x)^3}{39930 (3+5 x)^{3/2}}-\frac{511 (2+3 x)^4}{242 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^5}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{261331 \sqrt{1-2 x} (2+3 x)^2}{2196150 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (190406711+78981180 x)}{117128000}+\frac{753543 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8000 \sqrt{10}}\\ \end{align*}

Mathematica [C]  time = 5.3143, size = 312, normalized size = 1.82 \[ \frac{239 \left (2555520000 (1-2 x)^{7/2} (3 x+2)^4 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},2,2,2,\frac{7}{2}\right \},\left \{1,1,1,\frac{9}{2}\right \},\frac{5}{11} (1-2 x)\right )[phone] (1-2 x)^{5/2} \left (6 x^2+x-2\right )^3 \, _2F_1\left (\frac{3}{2},\frac{11}{2};\frac{13}{2};\frac{5}{11} (1-2 x)\right )+847 \sqrt{55} \left (\sqrt{10-20 x} \sqrt{5 x+3} \left (104976000 x^7+31298400 x^6-23823180 x^5-179946603 x^4+114920076 x^3+695191648 x^2+1209328624 x+353337912\right )+43923 \left (19521 x^4-40932 x^3-387936 x^2-241968 x-62504\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )\right )}{2551047840000 \sqrt{22} (1-2 x)^4}-\frac{259 \left (10 \left (-3234330 x^3+6746215 x^2+11581424 x+3821563\right )+2479653 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{106480000 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{3 (3 x+2)^5}{20 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(-3*(2 + 3*x)^5)/(20*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (259*(10*(3821563 + 11581424*x + 6746215*x^2 - 3234330
*x^3) + 2479653*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]))/(106480000*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2)) + (239*(847*Sqrt[55]*(Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*(353337912 + 1209328624*x + 695191648*x^2 +
114920076*x^3 - 179946603*x^4 - 23823180*x^5 + 31298400*x^6 + 104976000*x^7) + 43923*(-62504 - 241968*x - 387
936*x^2 - 40932*x^3 + 19521*x^4)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]) + 1176000000*(1 - 2*x)^(5/2)*(-2 + x + 6*x^
2)^3*Hypergeometric2F1[3/2, 11/2, 13/2, (5*(1 - 2*x))/11] + 2555520000*(1 - 2*x)^(7/2)*(2 + 3*x)^4*Hypergeomet
ricPFQ[{-1/2, 2, 2, 2, 7/2}, {1, 1, 1, 9/2}, (5*(1 - 2*x))/11]))/(2551047840000*Sqrt[22]*(1 - 2*x)^4)

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Maple [A]  time = 0.015, size = 199, normalized size = 1.2 \begin{align*}{\frac{1}{7027680000\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 3309786918900\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}-256158936000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-1959615860400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1952774282151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-198587215134\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5482566715380\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -398641171080\,x\sqrt{-10\,{x}^{2}-x+3}-888742129180\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/7027680000*(1-2*x)^(1/2)*(3309786918900*10^(1/2)*arcsin(20/11*x+1/11)*x^4-256158936000*x^5*(-10*x^2-x+3)^(1/
2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^3-1959615860400*x^4*(-10*x^2-x+3)^(1/2)-1952774282151*10^(1/2)
*arcsin(20/11*x+1/11)*x^2+5046848711200*x^3*(-10*x^2-x+3)^(1/2)-198587215134*10^(1/2)*arcsin(20/11*x+1/11)*x+5
482566715380*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-398641171080*x*(-10*x^2-x+3)^(
1/2)-888742129180*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 2.50405, size = 289, normalized size = 1.69 \begin{align*} -\frac{729 \, x^{5}}{20 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{111537 \, x^{4}}{400 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{251181}{234256000} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{753543}{160000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{90676341}{117128000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{170985889 \, x}{7027680 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{766611 \, x^{2}}{1000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1005653687}{878460000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{416356591 \, x}{3630000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{496819753}{3630000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-729/20*x^5/(-10*x^2 - x + 3)^(3/2) - 111537/400*x^4/(-10*x^2 - x + 3)^(3/2) + 251181/234256000*x*(7220*x/sqrt
(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x + 3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)
^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 753543/160000*sqrt(10)*arcsin(-20/11*x - 1/11) + 90676341/117128000*
sqrt(-10*x^2 - x + 3) - 170985889/7027680*x/sqrt(-10*x^2 - x + 3) + 766611/1000*x^2/(-10*x^2 - x + 3)^(3/2) +
1005653687/878460000/sqrt(-10*x^2 - x + 3) + 416356591/3630000*x/(-10*x^2 - x + 3)^(3/2) - 496819753/3630000/(
-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.85965, size = 447, normalized size = 2.61 \begin{align*} -\frac{33097869189 \, \sqrt{10}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{7027680000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/7027680000*(33097869189*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt
(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(12807946800*x^5 + 97980793020*x^4 - 252342435560*x^3 - 274128
335769*x^2 + 19932058554*x + 44437106459)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 2.56209, size = 282, normalized size = 1.65 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{2196150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{753543}{80000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{37 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{16637500 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (32019867 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 93 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 110347010662 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{219615000000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{1221 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{137259375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/2196150000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 753543/80000*sqrt(10)*arcsin(1
/11*sqrt(22)*sqrt(5*x + 3)) - 37/16637500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/2196
15000000*(4*(32019867*(4*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) - 110347010662*sqrt(5))*(5*x + 3) + 1820310
410259*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/137259375*(1221*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3