3.88 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

[Out]

(-2*A*(b*x + c*x^2)^(5/2))/(7*b*x^6) - (2*(7*b*B - 2*A*c)*(b*x + c*x^2)^(5/2))/(
35*b^2*x^5)

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Rubi [A]  time = 0.131583, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^5}-\frac{2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^6,x]

[Out]

(-2*A*(b*x + c*x^2)^(5/2))/(7*b*x^6) - (2*(7*b*B - 2*A*c)*(b*x + c*x^2)^(5/2))/(
35*b^2*x^5)

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Rubi in Sympy [A]  time = 8.17132, size = 53, normalized size = 0.93 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 b x^{6}} + \frac{4 \left (A c - \frac{7 B b}{2}\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{35 b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**6,x)

[Out]

-2*A*(b*x + c*x**2)**(5/2)/(7*b*x**6) + 4*(A*c - 7*B*b/2)*(b*x + c*x**2)**(5/2)/
(35*b**2*x**5)

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Mathematica [A]  time = 0.0775476, size = 36, normalized size = 0.63 \[ -\frac{2 (x (b+c x))^{5/2} (5 A b-2 A c x+7 b B x)}{35 b^2 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^6,x]

[Out]

(-2*(x*(b + c*x))^(5/2)*(5*A*b + 7*b*B*x - 2*A*c*x))/(35*b^2*x^6)

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Maple [A]  time = 0.008, size = 40, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,Acx+7\,xBb+5\,Ab \right ) }{35\,{x}^{5}{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^6,x)

[Out]

-2/35*(c*x+b)*(-2*A*c*x+7*B*b*x+5*A*b)*(c*x^2+b*x)^(3/2)/x^5/b^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((c*x^2 + b*x)^(3/2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282935, size = 105, normalized size = 1.84 \[ -\frac{2 \,{\left (5 \, A b^{3} +{\left (7 \, B b c^{2} - 2 \, A c^{3}\right )} x^{3} +{\left (14 \, B b^{2} c + A b c^{2}\right )} x^{2} +{\left (7 \, B b^{3} + 8 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{35 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

-2/35*(5*A*b^3 + (7*B*b*c^2 - 2*A*c^3)*x^3 + (14*B*b^2*c + A*b*c^2)*x^2 + (7*B*b
^3 + 8*A*b^2*c)*x)*sqrt(c*x^2 + b*x)/(b^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**6,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/x**6, x)

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GIAC/XCAS [A]  time = 0.279513, size = 420, normalized size = 7.37 \[ \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B c^{2} + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b c^{\frac{3}{2}} + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A c^{\frac{5}{2}} + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{2} c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b c^{2} + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{3} \sqrt{c} + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac{3}{2}} + 7 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{4} + 98 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{3} c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{4} \sqrt{c} + 5 \, A b^{5}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

2/35*(35*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^2 + 70*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^5*B*b*c^(3/2) + 35*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^(5/2) + 70*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*
c^2 + 35*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*sqrt(c) + 140*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^3*A*b^2*c^(3/2) + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4 + 98*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c + 35*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A
*b^4*sqrt(c) + 5*A*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^7