3.21 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=74 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{315 b^3 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(9*b*x^7) + (8*c*(b*x + c*x^2)^(5/2))/(63*b^2*x^6) - (1
6*c^2*(b*x + c*x^2)^(5/2))/(315*b^3*x^5)

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Rubi [A]  time = 0.0936088, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{5/2}}{315 b^3 x^5}+\frac{8 c \left (b x+c x^2\right )^{5/2}}{63 b^2 x^6}-\frac{2 \left (b x+c x^2\right )^{5/2}}{9 b x^7} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^7,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(9*b*x^7) + (8*c*(b*x + c*x^2)^(5/2))/(63*b^2*x^6) - (1
6*c^2*(b*x + c*x^2)^(5/2))/(315*b^3*x^5)

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Rubi in Sympy [A]  time = 9.38063, size = 68, normalized size = 0.92 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{9 b x^{7}} + \frac{8 c \left (b x + c x^{2}\right )^{\frac{5}{2}}}{63 b^{2} x^{6}} - \frac{16 c^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{315 b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**7,x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(9*b*x**7) + 8*c*(b*x + c*x**2)**(5/2)/(63*b**2*x**6) -
16*c**2*(b*x + c*x**2)**(5/2)/(315*b**3*x**5)

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Mathematica [A]  time = 0.0412509, size = 40, normalized size = 0.54 \[ -\frac{2 (x (b+c x))^{5/2} \left (35 b^2-20 b c x+8 c^2 x^2\right )}{315 b^3 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^7,x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(35*b^2 - 20*b*c*x + 8*c^2*x^2))/(315*b^3*x^7)

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-20\,bcx+35\,{b}^{2} \right ) }{315\,{x}^{6}{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^7,x)

[Out]

-2/315*(c*x+b)*(8*c^2*x^2-20*b*c*x+35*b^2)*(c*x^2+b*x)^(3/2)/x^6/b^3

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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((c*x^2 + b*x)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216181, size = 81, normalized size = 1.09 \[ -\frac{2 \,{\left (8 \, c^{4} x^{4} - 4 \, b c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + 50 \, b^{3} c x + 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^7,x, algorithm="fricas")

[Out]

-2/315*(8*c^4*x^4 - 4*b*c^3*x^3 + 3*b^2*c^2*x^2 + 50*b^3*c*x + 35*b^4)*sqrt(c*x^
2 + b*x)/(b^3*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**7,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.21575, size = 262, normalized size = 3.54 \[ \frac{2 \,{\left (420 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} c^{3} + 1575 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} b c^{\frac{5}{2}} + 2583 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b^{2} c^{2} + 2310 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{3} c^{\frac{3}{2}} + 1170 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{4} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{5} \sqrt{c} + 35 \, b^{6}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^7,x, algorithm="giac")

[Out]

2/315*(420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*c^3 + 1575*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*b*c^(5/2) + 2583*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^2*c^2 + 2310*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^3*b^3*c^(3/2) + 1170*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^2*b^4*c + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^5*sqrt(c) + 35*b^6)/(sqrt(c)*x
- sqrt(c*x^2 + b*x))^9