3.33 \(\int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(7*a*x^7)

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Rubi [A]  time = 0.0292503, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(7*a*x^7)

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Rubi in Sympy [A]  time = 3.63817, size = 20, normalized size = 0.87 \[ - \frac{2 \left (a x + b x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**7,x)

[Out]

-2*(a*x + b*x**2)**(7/2)/(7*a*x**7)

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Mathematica [A]  time = 0.0275825, size = 28, normalized size = 1.22 \[ -\frac{2 (a+b x)^3 \sqrt{x (a+b x)}}{7 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^2)^(5/2)/x^7,x]

[Out]

(-2*(a + b*x)^3*Sqrt[x*(a + b*x)])/(7*a*x^4)

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Maple [A]  time = 0.005, size = 25, normalized size = 1.1 \[ -{\frac{2\,bx+2\,a}{7\,{x}^{6}a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^7,x)

[Out]

-2/7/x^6*(b*x+a)/a*(b*x^2+a*x)^(5/2)

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230982, size = 62, normalized size = 2.7 \[ -\frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x}}{7 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x^2 + a*x)/(a*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**7,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.216547, size = 259, normalized size = 11.26 \[ \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} b^{3} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a b^{\frac{5}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{3} b^{\frac{3}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{4} b + 7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{5} \sqrt{b} + a^{6}\right )}}{7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^7,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3 + 21*(sqrt(b)*x - sqrt(b*x^2 + a*x)
)^5*a*b^(5/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2*b^2 + 35*(sqrt(b)*x - s
qrt(b*x^2 + a*x))^3*a^3*b^(3/2) + 21*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^4*b + 7
*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt(b) + a^6)/(sqrt(b)*x - sqrt(b*x^2 + a*
x))^7