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

Optimal. Leaf size=118 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]

[Out]

(5*a^4*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(512*b^3) - (5*a^2*(a + 2*b*x)*(a*x + b*x^
2)^(3/2))/(192*b^2) + ((a + 2*b*x)*(a*x + b*x^2)^(5/2))/(12*b) - (5*a^6*ArcTanh[
(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(512*b^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0954816, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2),x]

[Out]

(5*a^4*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(512*b^3) - (5*a^2*(a + 2*b*x)*(a*x + b*x^
2)^(3/2))/(192*b^2) + ((a + 2*b*x)*(a*x + b*x^2)^(5/2))/(12*b) - (5*a^6*ArcTanh[
(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(512*b^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.2079, size = 110, normalized size = 0.93 \[ - \frac{5 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{512 b^{\frac{7}{2}}} + \frac{5 a^{4} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{512 b^{3}} - \frac{5 a^{2} \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{3}{2}}}{192 b^{2}} + \frac{\left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{5}{2}}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**6*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(512*b**(7/2)) + 5*a**4*(a + 2*b*x)*
sqrt(a*x + b*x**2)/(512*b**3) - 5*a**2*(a + 2*b*x)*(a*x + b*x**2)**(3/2)/(192*b*
*2) + (a + 2*b*x)*(a*x + b*x**2)**(5/2)/(12*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.138311, size = 122, normalized size = 1.03 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-\frac{15 a^6 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}\right )}{1536 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(15*a^5 - 10*a^4*b*x + 8*a^3*b^2*x^2 + 432*a^2*b^3*x
^3 + 640*a*b^4*x^4 + 256*b^5*x^5) - (15*a^6*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x
]])/(Sqrt[x]*Sqrt[a + b*x])))/(1536*b^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 134, normalized size = 1.1 \[{\frac{2\,bx+a}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}x}{96\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2),x)

[Out]

1/12*(2*b*x+a)*(b*x^2+a*x)^(5/2)/b-5/96/b*a^2*(b*x^2+a*x)^(3/2)*x-5/192/b^2*a^3*
(b*x^2+a*x)^(3/2)+5/256/b^2*a^4*(b*x^2+a*x)^(1/2)*x+5/512/b^3*a^5*(b*x^2+a*x)^(1
/2)-5/1024/b^(7/2)*a^6*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))

_______________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231287, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{3072 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{6} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{1536 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3072*(15*a^6*log((2*b*x + a)*sqrt(b) - 2*sqrt(b*x^2 + a*x)*b) + 2*(256*b^5*x^
5 + 640*a*b^4*x^4 + 432*a^2*b^3*x^3 + 8*a^3*b^2*x^2 - 10*a^4*b*x + 15*a^5)*sqrt(
b*x^2 + a*x)*sqrt(b))/b^(7/2), -1/1536*(15*a^6*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)
/(b*x)) - (256*b^5*x^5 + 640*a*b^4*x^4 + 432*a^2*b^3*x^3 + 8*a^3*b^2*x^2 - 10*a^
4*b*x + 15*a^5)*sqrt(b*x^2 + a*x)*sqrt(-b))/(sqrt(-b)*b^3)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x + b*x**2)**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223985, size = 144, normalized size = 1.22 \[ \frac{5 \, a^{6}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{1024 \, b^{\frac{7}{2}}} + \frac{1}{1536} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{5}}{b^{3}} - 2 \,{\left (\frac{5 \, a^{4}}{b^{2}} - 4 \,{\left (\frac{a^{3}}{b} + 2 \,{\left (27 \, a^{2} + 8 \,{\left (2 \, b^{2} x + 5 \, a b\right )} x\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/1024*a^6*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(7/2) + 1/1
536*sqrt(b*x^2 + a*x)*(15*a^5/b^3 - 2*(5*a^4/b^2 - 4*(a^3/b + 2*(27*a^2 + 8*(2*b
^2*x + 5*a*b)*x)*x)*x)*x)