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

Optimal. Leaf size=61 \[ \frac{2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.134809, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 B \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.00698, size = 56, normalized size = 0.92 \[ \frac{2 B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 c x^{\frac{3}{2}}} + \frac{4 \left (\frac{7 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{35 c^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0596871, size = 37, normalized size = 0.61 \[ \frac{2 (x (b+c x))^{5/2} (7 A c-2 b B+5 B c x)}{35 c^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 5\,Bcx+7\,Ac-2\,Bb \right ) }{35\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.711498, size = 174, normalized size = 2.85 \[ \frac{2 \,{\left (5 \, b c x^{2} + 5 \, b^{2} x +{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} x\right )} \sqrt{c x + b} A}{15 \, c x} + \frac{2 \,{\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b} B}{105 \, c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*b*c*x^2 + 5*b^2*x + (3*c^2*x^2 + b*c*x - 2*b^2)*x)*sqrt(c*x + b)*A/(c*x)
+ 2/105*((15*c^3*x^3 + 3*b*c^2*x^2 - 4*b^2*c*x + 8*b^3)*x^2 + 7*(3*b*c^2*x^3 +
b^2*c*x^2 - 2*b^3*x)*x)*sqrt(c*x + b)*B/(c^2*x^2)

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Fricas [A]  time = 0.272104, size = 144, normalized size = 2.36 \[ \frac{2 \,{\left (5 \, B c^{4} x^{5} +{\left (13 \, B b c^{3} + 7 \, A c^{4}\right )} x^{4} + 3 \,{\left (3 \, B b^{2} c^{2} + 7 \, A b c^{3}\right )} x^{3} -{\left (B b^{3} c - 21 \, A b^{2} c^{2}\right )} x^{2} -{\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*B*c^4*x^5 + (13*B*b*c^3 + 7*A*c^4)*x^4 + 3*(3*B*b^2*c^2 + 7*A*b*c^3)*x^3
- (B*b^3*c - 21*A*b^2*c^2)*x^2 - (2*B*b^4 - 7*A*b^3*c)*x)/(sqrt(c*x^2 + b*x)*c^
2*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.276828, size = 201, normalized size = 3.3 \[ -\frac{2}{105} \, B c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, B b{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{15} \, A c{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, A b{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*B*c*(8*b^(7/2)/c^3 - (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x
+ b)^(3/2)*b^2)/c^3) + 2/15*B*b*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x +
b)^(3/2)*b)/c^2) + 2/15*A*c*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(3
/2)*b)/c^2) + 2/3*A*b*((c*x + b)^(3/2)/c - b^(3/2)/c)