3.60 \(\int \frac{x^6}{\left (a x+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{4 b^{9/2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}} \]

[Out]

(-2*x^5)/(3*b*(a*x + b*x^2)^(3/2)) - (14*x^3)/(3*b^2*Sqrt[a*x + b*x^2]) - (35*a*
Sqrt[a*x + b*x^2])/(4*b^4) + (35*x*Sqrt[a*x + b*x^2])/(6*b^3) + (35*a^2*ArcTanh[
(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(9/2))

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Rubi [A]  time = 0.179722, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{4 b^{9/2}}-\frac{35 a \sqrt{a x+b x^2}}{4 b^4}+\frac{35 x \sqrt{a x+b x^2}}{6 b^3}-\frac{14 x^3}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^5}{3 b \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a*x + b*x^2)^(5/2),x]

[Out]

(-2*x^5)/(3*b*(a*x + b*x^2)^(3/2)) - (14*x^3)/(3*b^2*Sqrt[a*x + b*x^2]) - (35*a*
Sqrt[a*x + b*x^2])/(4*b^4) + (35*x*Sqrt[a*x + b*x^2])/(6*b^3) + (35*a^2*ArcTanh[
(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*b^(9/2))

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Rubi in Sympy [A]  time = 20.5545, size = 114, normalized size = 0.93 \[ \frac{35 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{4 b^{\frac{9}{2}}} - \frac{35 a \sqrt{a x + b x^{2}}}{4 b^{4}} - \frac{2 x^{5}}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{14 x^{3}}{3 b^{2} \sqrt{a x + b x^{2}}} + \frac{35 x \sqrt{a x + b x^{2}}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**2+a*x)**(5/2),x)

[Out]

35*a**2*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(4*b**(9/2)) - 35*a*sqrt(a*x + b*x**
2)/(4*b**4) - 2*x**5/(3*b*(a*x + b*x**2)**(3/2)) - 14*x**3/(3*b**2*sqrt(a*x + b*
x**2)) + 35*x*sqrt(a*x + b*x**2)/(6*b**3)

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Mathematica [A]  time = 0.103081, size = 102, normalized size = 0.84 \[ \frac{x \left (105 a^2 \sqrt{x} (a+b x)^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )+\sqrt{b} x \left (-105 a^3-140 a^2 b x-21 a b^2 x^2+6 b^3 x^3\right )\right )}{12 b^{9/2} (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a*x + b*x^2)^(5/2),x]

[Out]

(x*(Sqrt[b]*x*(-105*a^3 - 140*a^2*b*x - 21*a*b^2*x^2 + 6*b^3*x^3) + 105*a^2*Sqrt
[x]*(a + b*x)^(3/2)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]]))/(12*b^(9/2)*(x*(a +
b*x))^(3/2))

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Maple [A]  time = 0.009, size = 176, normalized size = 1.4 \[{\frac{{x}^{5}}{2\,b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,a{x}^{4}}{4\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{35\,{a}^{2}{x}^{3}}{24\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{x}^{2}{a}^{3}}{16\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{4}x}{48\,{b}^{5}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{245\,{a}^{2}x}{24\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{35\,{a}^{3}}{48\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\frac{35\,{a}^{2}}{8}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^2+a*x)^(5/2),x)

[Out]

1/2*x^5/b/(b*x^2+a*x)^(3/2)-7/4*a/b^2*x^4/(b*x^2+a*x)^(3/2)-35/24*a^2/b^3*x^3/(b
*x^2+a*x)^(3/2)+35/16*a^3/b^4*x^2/(b*x^2+a*x)^(3/2)+35/48*a^4/b^5/(b*x^2+a*x)^(3
/2)*x-245/24*a^2/b^4/(b*x^2+a*x)^(1/2)*x-35/48*a^3/b^5/(b*x^2+a*x)^(1/2)+35/8*a^
2/b^(9/2)*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(x^6/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236177, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (6 \, b^{3} x^{4} - 21 \, a b^{2} x^{3} - 140 \, a^{2} b x^{2} - 105 \, a^{3} x\right )} \sqrt{b}}{24 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}, \frac{105 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (6 \, b^{3} x^{4} - 21 \, a b^{2} x^{3} - 140 \, a^{2} b x^{2} - 105 \, a^{3} x\right )} \sqrt{-b}}{12 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(105*(a^2*b*x + a^3)*sqrt(b*x^2 + a*x)*log((2*b*x + a)*sqrt(b) + 2*sqrt(b*
x^2 + a*x)*b) + 2*(6*b^3*x^4 - 21*a*b^2*x^3 - 140*a^2*b*x^2 - 105*a^3*x)*sqrt(b)
)/((b^5*x + a*b^4)*sqrt(b*x^2 + a*x)*sqrt(b)), 1/12*(105*(a^2*b*x + a^3)*sqrt(b*
x^2 + a*x)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) + (6*b^3*x^4 - 21*a*b^2*x^3
- 140*a^2*b*x^2 - 105*a^3*x)*sqrt(-b))/((b^5*x + a*b^4)*sqrt(b*x^2 + a*x)*sqrt(-
b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x**6/(x*(a + b*x))**(5/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^2 + a*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError