3.155 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{x^{5/2}} \, dx\)

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

[Out]

2*A*b^2*Sqrt[x] + (2*b*(b*B + 2*A*c)*x^(3/2))/3 + (2*c*(2*b*B + A*c)*x^(5/2))/5
+ (2*B*c^2*x^(7/2))/7

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Rubi [A]  time = 0.0917004, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ 2 A b^2 \sqrt{x}+\frac{2}{5} c x^{5/2} (A c+2 b B)+\frac{2}{3} b x^{3/2} (2 A c+b B)+\frac{2}{7} B c^2 x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/x^(5/2),x]

[Out]

2*A*b^2*Sqrt[x] + (2*b*(b*B + 2*A*c)*x^(3/2))/3 + (2*c*(2*b*B + A*c)*x^(5/2))/5
+ (2*B*c^2*x^(7/2))/7

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Rubi in Sympy [A]  time = 10.2289, size = 61, normalized size = 1. \[ 2 A b^{2} \sqrt{x} + \frac{2 B c^{2} x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{3}{2}} \left (2 A c + B b\right )}{3} + \frac{2 c x^{\frac{5}{2}} \left (A c + 2 B b\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**(5/2),x)

[Out]

2*A*b**2*sqrt(x) + 2*B*c**2*x**(7/2)/7 + 2*b*x**(3/2)*(2*A*c + B*b)/3 + 2*c*x**(
5/2)*(A*c + 2*B*b)/5

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Mathematica [A]  time = 0.0341499, size = 51, normalized size = 0.84 \[ \frac{2}{105} \sqrt{x} \left (105 A b^2+21 c x^2 (A c+2 b B)+35 b x (2 A c+b B)+15 B c^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^(5/2),x]

[Out]

(2*Sqrt[x]*(105*A*b^2 + 35*b*(b*B + 2*A*c)*x + 21*c*(2*b*B + A*c)*x^2 + 15*B*c^2
*x^3))/105

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Maple [A]  time = 0.01, size = 52, normalized size = 0.9 \[{\frac{30\,B{c}^{2}{x}^{3}+42\,A{c}^{2}{x}^{2}+84\,B{x}^{2}bc+140\,Abcx+70\,{b}^{2}Bx+210\,{b}^{2}A}{105}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x^(5/2),x)

[Out]

2/105*x^(1/2)*(15*B*c^2*x^3+21*A*c^2*x^2+42*B*b*c*x^2+70*A*b*c*x+35*B*b^2*x+105*
A*b^2)

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Maxima [A]  time = 0.683037, size = 69, normalized size = 1.13 \[ \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + 2 \, A b^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.284575, size = 69, normalized size = 1.13 \[ \frac{2}{105} \,{\left (15 \, B c^{2} x^{3} + 105 \, A b^{2} + 21 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 35 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*c^2*x^3 + 105*A*b^2 + 21*(2*B*b*c + A*c^2)*x^2 + 35*(B*b^2 + 2*A*b*c
)*x)*sqrt(x)

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Sympy [A]  time = 4.73507, size = 78, normalized size = 1.28 \[ 2 A b^{2} \sqrt{x} + \frac{4 A b c x^{\frac{3}{2}}}{3} + \frac{2 A c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{3}{2}}}{3} + \frac{4 B b c x^{\frac{5}{2}}}{5} + \frac{2 B c^{2} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/x**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.268248, size = 72, normalized size = 1.18 \[ \frac{2}{7} \, B c^{2} x^{\frac{7}{2}} + \frac{4}{5} \, B b c x^{\frac{5}{2}} + \frac{2}{5} \, A c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A b c x^{\frac{3}{2}} + 2 \, A b^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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