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

Optimal. Leaf size=139 \[ \frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b} \]

[Out]

(-5*a^5*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(1024*b^4) + (5*a^3*(a + 2*b*x)*(a*x + b*
x^2)^(3/2))/(384*b^3) - (a*(a + 2*b*x)*(a*x + b*x^2)^(5/2))/(24*b^2) + (a*x + b*
x^2)^(7/2)/(7*b) + (5*a^7*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(1024*b^(9/2))

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Rubi [A]  time = 0.138513, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{1024 b^{9/2}}-\frac{5 a^5 (a+2 b x) \sqrt{a x+b x^2}}{1024 b^4}+\frac{5 a^3 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{384 b^3}-\frac{a (a+2 b x) \left (a x+b x^2\right )^{5/2}}{24 b^2}+\frac{\left (a x+b x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x*(a*x + b*x^2)^(5/2),x]

[Out]

(-5*a^5*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(1024*b^4) + (5*a^3*(a + 2*b*x)*(a*x + b*
x^2)^(3/2))/(384*b^3) - (a*(a + 2*b*x)*(a*x + b*x^2)^(5/2))/(24*b^2) + (a*x + b*
x^2)^(7/2)/(7*b) + (5*a^7*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(1024*b^(9/2))

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Rubi in Sympy [A]  time = 17.6056, size = 129, normalized size = 0.93 \[ \frac{5 a^{7} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{1024 b^{\frac{9}{2}}} - \frac{5 a^{5} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{1024 b^{4}} + \frac{5 a^{3} \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{3}{2}}}{384 b^{3}} - \frac{a \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{5}{2}}}{24 b^{2}} + \frac{\left (a x + b x^{2}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**7*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(1024*b**(9/2)) - 5*a**5*(a + 2*b*x)*
sqrt(a*x + b*x**2)/(1024*b**4) + 5*a**3*(a + 2*b*x)*(a*x + b*x**2)**(3/2)/(384*b
**3) - a*(a + 2*b*x)*(a*x + b*x**2)**(5/2)/(24*b**2) + (a*x + b*x**2)**(7/2)/(7*
b)

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Mathematica [A]  time = 0.162649, size = 133, normalized size = 0.96 \[ \frac{\sqrt{x (a+b x)} \left (\frac{105 a^7 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}+\sqrt{b} \left (-105 a^6+70 a^5 b x-56 a^4 b^2 x^2+48 a^3 b^3 x^3+4736 a^2 b^4 x^4+7424 a b^5 x^5+3072 b^6 x^6\right )\right )}{21504 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(-105*a^6 + 70*a^5*b*x - 56*a^4*b^2*x^2 + 48*a^3*b^3
*x^3 + 4736*a^2*b^4*x^4 + 7424*a*b^5*x^5 + 3072*b^6*x^6) + (105*a^7*Log[b*Sqrt[x
] + Sqrt[b]*Sqrt[a + b*x]])/(Sqrt[x]*Sqrt[a + b*x])))/(21504*b^(9/2))

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Maple [A]  time = 0.007, size = 165, normalized size = 1.2 \[{\frac{1}{7\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{ax}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{24\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{3}x}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}}{384\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{5}x}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024\,{b}^{4}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{7}}{2048}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a*x)^(5/2),x)

[Out]

1/7*(b*x^2+a*x)^(7/2)/b-1/12*a/b*x*(b*x^2+a*x)^(5/2)-1/24*a^2/b^2*(b*x^2+a*x)^(5
/2)+5/192*a^3/b^2*(b*x^2+a*x)^(3/2)*x+5/384*a^4/b^3*(b*x^2+a*x)^(3/2)-5/512*a^5/
b^3*(b*x^2+a*x)^(1/2)*x-5/1024*a^6/b^4*(b*x^2+a*x)^(1/2)+5/2048*a^7/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((b*x^2 + a*x)^(5/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231603, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, a^{7} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (3072 \, b^{6} x^{6} + 7424 \, a b^{5} x^{5} + 4736 \, a^{2} b^{4} x^{4} + 48 \, a^{3} b^{3} x^{3} - 56 \, a^{4} b^{2} x^{2} + 70 \, a^{5} b x - 105 \, a^{6}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{43008 \, b^{\frac{9}{2}}}, \frac{105 \, a^{7} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (3072 \, b^{6} x^{6} + 7424 \, a b^{5} x^{5} + 4736 \, a^{2} b^{4} x^{4} + 48 \, a^{3} b^{3} x^{3} - 56 \, a^{4} b^{2} x^{2} + 70 \, a^{5} b x - 105 \, a^{6}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{21504 \, \sqrt{-b} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/43008*(105*a^7*log((2*b*x + a)*sqrt(b) + 2*sqrt(b*x^2 + a*x)*b) + 2*(3072*b^6
*x^6 + 7424*a*b^5*x^5 + 4736*a^2*b^4*x^4 + 48*a^3*b^3*x^3 - 56*a^4*b^2*x^2 + 70*
a^5*b*x - 105*a^6)*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2), 1/21504*(105*a^7*arctan(s
qrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) + (3072*b^6*x^6 + 7424*a*b^5*x^5 + 4736*a^2*b^4
*x^4 + 48*a^3*b^3*x^3 - 56*a^4*b^2*x^2 + 70*a^5*b*x - 105*a^6)*sqrt(b*x^2 + a*x)
*sqrt(-b))/(sqrt(-b)*b^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(x*(a + b*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.224097, size = 162, normalized size = 1.17 \[ -\frac{5 \, a^{7}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{2048 \, b^{\frac{9}{2}}} - \frac{1}{21504} \, \sqrt{b x^{2} + a x}{\left (\frac{105 \, a^{6}}{b^{4}} - 2 \,{\left (\frac{35 \, a^{5}}{b^{3}} - 4 \,{\left (\frac{7 \, a^{4}}{b^{2}} - 2 \,{\left (\frac{3 \, a^{3}}{b} + 8 \,{\left (37 \, a^{2} + 2 \,{\left (12 \, b^{2} x + 29 \, a b\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2048*a^7*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(9/2) - 1/
21504*sqrt(b*x^2 + a*x)*(105*a^6/b^4 - 2*(35*a^5/b^3 - 4*(7*a^4/b^2 - 2*(3*a^3/b
+ 8*(37*a^2 + 2*(12*b^2*x + 29*a*b)*x)*x)*x)*x)*x)