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

Optimal. Leaf size=164 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{5 a^5 \sqrt{x} \sqrt{a+b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^3 x^{5/2} \sqrt{a+b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a+b x}+\frac{1}{12} a x^{7/2} (a+b x)^{3/2}+\frac{1}{6} x^{7/2} (a+b x)^{5/2} \]

[Out]

(5*a^5*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a + b*x])/(768*b^2
) + (a^3*x^(5/2)*Sqrt[a + b*x])/(192*b) + (a^2*x^(7/2)*Sqrt[a + b*x])/32 + (a*x^
(7/2)*(a + b*x)^(3/2))/12 + (x^(7/2)*(a + b*x)^(5/2))/6 - (5*a^6*ArcTanh[(Sqrt[b
]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

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Rubi [A]  time = 0.149244, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{512 b^{7/2}}+\frac{5 a^5 \sqrt{x} \sqrt{a+b x}}{512 b^3}-\frac{5 a^4 x^{3/2} \sqrt{a+b x}}{768 b^2}+\frac{a^3 x^{5/2} \sqrt{a+b x}}{192 b}+\frac{1}{32} a^2 x^{7/2} \sqrt{a+b x}+\frac{1}{12} a x^{7/2} (a+b x)^{3/2}+\frac{1}{6} x^{7/2} (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x)^(5/2),x]

[Out]

(5*a^5*Sqrt[x]*Sqrt[a + b*x])/(512*b^3) - (5*a^4*x^(3/2)*Sqrt[a + b*x])/(768*b^2
) + (a^3*x^(5/2)*Sqrt[a + b*x])/(192*b) + (a^2*x^(7/2)*Sqrt[a + b*x])/32 + (a*x^
(7/2)*(a + b*x)^(3/2))/12 + (x^(7/2)*(a + b*x)^(5/2))/6 - (5*a^6*ArcTanh[(Sqrt[b
]*Sqrt[x])/Sqrt[a + b*x]])/(512*b^(7/2))

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Rubi in Sympy [A]  time = 24.118, size = 160, normalized size = 0.98 \[ - \frac{5 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{512 b^{\frac{7}{2}}} - \frac{5 a^{5} \sqrt{x} \sqrt{a + b x}}{512 b^{3}} - \frac{5 a^{4} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{768 b^{3}} - \frac{a^{3} \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{192 b^{3}} + \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{32 b^{3}} - \frac{a x^{\frac{3}{2}} \left (a + b x\right )^{\frac{7}{2}}}{12 b^{2}} + \frac{x^{\frac{5}{2}} \left (a + b x\right )^{\frac{7}{2}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**6*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(512*b**(7/2)) - 5*a**5*sqrt(x)*sqr
t(a + b*x)/(512*b**3) - 5*a**4*sqrt(x)*(a + b*x)**(3/2)/(768*b**3) - a**3*sqrt(x
)*(a + b*x)**(5/2)/(192*b**3) + a**2*sqrt(x)*(a + b*x)**(7/2)/(32*b**3) - a*x**(
3/2)*(a + b*x)**(7/2)/(12*b**2) + x**(5/2)*(a + b*x)**(7/2)/(6*b)

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Mathematica [A]  time = 0.0963994, size = 111, normalized size = 0.68 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^5-10 a^4 b x+8 a^3 b^2 x^2+432 a^2 b^3 x^3+640 a b^4 x^4+256 b^5 x^5\right )-15 a^6 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{1536 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(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]])/(1536*b^(7/2))

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Maple [A]  time = 0.007, size = 156, normalized size = 1. \[{\frac{1}{6\,b}{x}^{{\frac{5}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{32\,{b}^{3}}\sqrt{x} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{3}}{192\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{5\,{a}^{4}}{768\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{6}}{1024}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)^(5/2),x)

[Out]

1/6/b*x^(5/2)*(b*x+a)^(7/2)-1/12*a/b^2*x^(3/2)*(b*x+a)^(7/2)+1/32*a^2/b^3*x^(1/2
)*(b*x+a)^(7/2)-1/192*a^3/b^3*(b*x+a)^(5/2)*x^(1/2)-5/768*a^4/b^3*(b*x+a)^(3/2)*
x^(1/2)-5/512*a^5*x^(1/2)*(b*x+a)^(1/2)/b^3-5/1024*a^6/b^(7/2)*(x*(b*x+a))^(1/2)
/(b*x+a)^(1/2)/x^(1/2)*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 + a)^(5/2)*x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2262, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{6} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{3072 \, b^{\frac{7}{2}}}, -\frac{15 \, a^{6} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (256 \, b^{5} x^{5} + 640 \, a b^{4} x^{4} + 432 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} - 10 \, a^{4} b x + 15 \, a^{5}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{1536 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3072*(15*a^6*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(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 + a)*sqrt(b)*sqrt(x))/b^(7/2), -1/1536*(15*a^6*arctan(sqrt(b*x + a)*sqr
t(-b)/(b*sqrt(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 + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 24.8317, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x