3.24 \(\int x^2 \left (a x+b x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=163 \[ -\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{16384 b^{11/2}}+\frac{45 a^6 (a+2 b x) \sqrt{a x+b x^2}}{16384 b^5}-\frac{15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac{3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac{9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac{x \left (a x+b x^2\right )^{7/2}}{8 b} \]

[Out]

(45*a^6*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(16384*b^5) - (15*a^4*(a + 2*b*x)*(a*x +
b*x^2)^(3/2))/(2048*b^4) + (3*a^2*(a + 2*b*x)*(a*x + b*x^2)^(5/2))/(128*b^3) - (
9*a*(a*x + b*x^2)^(7/2))/(112*b^2) + (x*(a*x + b*x^2)^(7/2))/(8*b) - (45*a^8*Arc
Tanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(16384*b^(11/2))

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Rubi [A]  time = 0.213576, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{45 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{16384 b^{11/2}}+\frac{45 a^6 (a+2 b x) \sqrt{a x+b x^2}}{16384 b^5}-\frac{15 a^4 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{2048 b^4}+\frac{3 a^2 (a+2 b x) \left (a x+b x^2\right )^{5/2}}{128 b^3}-\frac{9 a \left (a x+b x^2\right )^{7/2}}{112 b^2}+\frac{x \left (a x+b x^2\right )^{7/2}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a*x + b*x^2)^(5/2),x]

[Out]

(45*a^6*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(16384*b^5) - (15*a^4*(a + 2*b*x)*(a*x +
b*x^2)^(3/2))/(2048*b^4) + (3*a^2*(a + 2*b*x)*(a*x + b*x^2)^(5/2))/(128*b^3) - (
9*a*(a*x + b*x^2)^(7/2))/(112*b^2) + (x*(a*x + b*x^2)^(7/2))/(8*b) - (45*a^8*Arc
Tanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(16384*b^(11/2))

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Rubi in Sympy [A]  time = 24.816, size = 155, normalized size = 0.95 \[ - \frac{45 a^{8} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{16384 b^{\frac{11}{2}}} + \frac{45 a^{6} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{16384 b^{5}} - \frac{15 a^{4} \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{3}{2}}}{2048 b^{4}} + \frac{3 a^{2} \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{5}{2}}}{128 b^{3}} - \frac{9 a \left (a x + b x^{2}\right )^{\frac{7}{2}}}{112 b^{2}} + \frac{x \left (a x + b x^{2}\right )^{\frac{7}{2}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a*x)**(5/2),x)

[Out]

-45*a**8*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(16384*b**(11/2)) + 45*a**6*(a + 2*
b*x)*sqrt(a*x + b*x**2)/(16384*b**5) - 15*a**4*(a + 2*b*x)*(a*x + b*x**2)**(3/2)
/(2048*b**4) + 3*a**2*(a + 2*b*x)*(a*x + b*x**2)**(5/2)/(128*b**3) - 9*a*(a*x +
b*x**2)**(7/2)/(112*b**2) + x*(a*x + b*x**2)**(7/2)/(8*b)

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Mathematica [A]  time = 0.237787, size = 156, normalized size = 0.96 \[ \frac{\sqrt{b} x \left (315 a^8+105 a^7 b x-42 a^6 b^2 x^2+24 a^5 b^3 x^3-16 a^4 b^4 x^4+20864 a^3 b^5 x^5+54528 a^2 b^6 x^6+48128 a b^7 x^7+14336 b^8 x^8\right )-315 a^8 \sqrt{x} \sqrt{a+b x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{114688 b^{11/2} \sqrt{x (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a*x + b*x^2)^(5/2),x]

[Out]

(Sqrt[b]*x*(315*a^8 + 105*a^7*b*x - 42*a^6*b^2*x^2 + 24*a^5*b^3*x^3 - 16*a^4*b^4
*x^4 + 20864*a^3*b^5*x^5 + 54528*a^2*b^6*x^6 + 48128*a*b^7*x^7 + 14336*b^8*x^8)
- 315*a^8*Sqrt[x]*Sqrt[a + b*x]*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(114688*
b^(11/2)*Sqrt[x*(a + b*x)])

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Maple [A]  time = 0.009, size = 185, normalized size = 1.1 \[{\frac{x}{8\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{9\,a}{112\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{2}x}{64\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{3}}{128\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{a}^{4}x}{1024\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{a}^{5}}{2048\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{a}^{6}x}{8192\,{b}^{4}}\sqrt{b{x}^{2}+ax}}+{\frac{45\,{a}^{7}}{16384\,{b}^{5}}\sqrt{b{x}^{2}+ax}}-{\frac{45\,{a}^{8}}{32768}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a*x)^(5/2),x)

[Out]

1/8*x*(b*x^2+a*x)^(7/2)/b-9/112*a*(b*x^2+a*x)^(7/2)/b^2+3/64*a^2/b^2*(b*x^2+a*x)
^(5/2)*x+3/128*a^3/b^3*(b*x^2+a*x)^(5/2)-15/1024*a^4/b^3*(b*x^2+a*x)^(3/2)*x-15/
2048*a^5/b^4*(b*x^2+a*x)^(3/2)+45/8192*a^6/b^4*(b*x^2+a*x)^(1/2)*x+45/16384*a^7/
b^5*(b*x^2+a*x)^(1/2)-45/32768*a^8/b^(11/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(
1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232415, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{8} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (14336 \, b^{7} x^{7} + 33792 \, a b^{6} x^{6} + 20736 \, a^{2} b^{5} x^{5} + 128 \, a^{3} b^{4} x^{4} - 144 \, a^{4} b^{3} x^{3} + 168 \, a^{5} b^{2} x^{2} - 210 \, a^{6} b x + 315 \, a^{7}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{229376 \, b^{\frac{11}{2}}}, -\frac{315 \, a^{8} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (14336 \, b^{7} x^{7} + 33792 \, a b^{6} x^{6} + 20736 \, a^{2} b^{5} x^{5} + 128 \, a^{3} b^{4} x^{4} - 144 \, a^{4} b^{3} x^{3} + 168 \, a^{5} b^{2} x^{2} - 210 \, a^{6} b x + 315 \, a^{7}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{114688 \, \sqrt{-b} b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)*x^2,x, algorithm="fricas")

[Out]

[1/229376*(315*a^8*log((2*b*x + a)*sqrt(b) - 2*sqrt(b*x^2 + a*x)*b) + 2*(14336*b
^7*x^7 + 33792*a*b^6*x^6 + 20736*a^2*b^5*x^5 + 128*a^3*b^4*x^4 - 144*a^4*b^3*x^3
+ 168*a^5*b^2*x^2 - 210*a^6*b*x + 315*a^7)*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2),
-1/114688*(315*a^8*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) - (14336*b^7*x^7 +
33792*a*b^6*x^6 + 20736*a^2*b^5*x^5 + 128*a^3*b^4*x^4 - 144*a^4*b^3*x^3 + 168*a^
5*b^2*x^2 - 210*a^6*b*x + 315*a^7)*sqrt(b*x^2 + a*x)*sqrt(-b))/(sqrt(-b)*b^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (x \left (a + b x\right )\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x**2*(x*(a + b*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.224625, size = 177, normalized size = 1.09 \[ \frac{45 \, a^{8}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{32768 \, b^{\frac{11}{2}}} + \frac{1}{114688} \, \sqrt{b x^{2} + a x}{\left (\frac{315 \, a^{7}}{b^{5}} - 2 \,{\left (\frac{105 \, a^{6}}{b^{4}} - 4 \,{\left (\frac{21 \, a^{5}}{b^{3}} - 2 \,{\left (\frac{9 \, a^{4}}{b^{2}} - 8 \,{\left (\frac{a^{3}}{b} + 2 \,{\left (81 \, a^{2} + 4 \,{\left (14 \, b^{2} x + 33 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)*x^2,x, algorithm="giac")

[Out]

45/32768*a^8*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(11/2) +
1/114688*sqrt(b*x^2 + a*x)*(315*a^7/b^5 - 2*(105*a^6/b^4 - 4*(21*a^5/b^3 - 2*(9*
a^4/b^2 - 8*(a^3/b + 2*(81*a^2 + 4*(14*b^2*x + 33*a*b)*x)*x)*x)*x)*x)*x)