3.164 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{5} b^2 x^{5/2} (3 A c+b B)+\frac{2}{3} A b^3 x^{3/2}+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{6}{7} b c x^{7/2} (A c+b B)+\frac{2}{11} B c^3 x^{11/2} \]

[Out]

(2*A*b^3*x^(3/2))/3 + (2*b^2*(b*B + 3*A*c)*x^(5/2))/5 + (6*b*c*(b*B + A*c)*x^(7/2))/7 + (2*c^2*(3*b*B + A*c)*x
^(9/2))/9 + (2*B*c^3*x^(11/2))/11

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Rubi [A]  time = 0.0438203, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {765} \[ \frac{2}{5} b^2 x^{5/2} (3 A c+b B)+\frac{2}{3} A b^3 x^{3/2}+\frac{2}{9} c^2 x^{9/2} (A c+3 b B)+\frac{6}{7} b c x^{7/2} (A c+b B)+\frac{2}{11} B c^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^3*x^(3/2))/3 + (2*b^2*(b*B + 3*A*c)*x^(5/2))/5 + (6*b*c*(b*B + A*c)*x^(7/2))/7 + (2*c^2*(3*b*B + A*c)*x
^(9/2))/9 + (2*B*c^3*x^(11/2))/11

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^3}{x^{5/2}} \, dx &=\int \left (A b^3 \sqrt{x}+b^2 (b B+3 A c) x^{3/2}+3 b c (b B+A c) x^{5/2}+c^2 (3 b B+A c) x^{7/2}+B c^3 x^{9/2}\right ) \, dx\\ &=\frac{2}{3} A b^3 x^{3/2}+\frac{2}{5} b^2 (b B+3 A c) x^{5/2}+\frac{6}{7} b c (b B+A c) x^{7/2}+\frac{2}{9} c^2 (3 b B+A c) x^{9/2}+\frac{2}{11} B c^3 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0430358, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{3/2} (b+c x)^4-\frac{1}{315} x^{3/2} \left (189 b^2 c x+105 b^3+135 b c^2 x^2+35 c^3 x^3\right ) (3 b B-11 A c)\right )}{11 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*x^(3/2)*(b + c*x)^4 - ((3*b*B - 11*A*c)*x^(3/2)*(105*b^3 + 189*b^2*c*x + 135*b*c^2*x^2 + 35*c^3*x^3))/31
5))/(11*c)

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Maple [A]  time = 0.006, size = 76, normalized size = 0.9 \begin{align*}{\frac{630\,B{c}^{3}{x}^{4}+770\,A{x}^{3}{c}^{3}+2310\,B{x}^{3}b{c}^{2}+2970\,A{x}^{2}b{c}^{2}+2970\,B{x}^{2}{b}^{2}c+4158\,A{b}^{2}cx+1386\,{b}^{3}Bx+2310\,A{b}^{3}}{3465}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x)

[Out]

2/3465*x^(3/2)*(315*B*c^3*x^4+385*A*c^3*x^3+1155*B*b*c^2*x^3+1485*A*b*c^2*x^2+1485*B*b^2*c*x^2+2079*A*b^2*c*x+
693*B*b^3*x+1155*A*b^3)

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Maxima [A]  time = 1.009, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + \frac{2}{9} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/11*B*c^3*x^(11/2) + 2/3*A*b^3*x^(3/2) + 2/9*(3*B*b*c^2 + A*c^3)*x^(9/2) + 6/7*(B*b^2*c + A*b*c^2)*x^(7/2) +
2/5*(B*b^3 + 3*A*b^2*c)*x^(5/2)

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Fricas [A]  time = 1.75289, size = 186, normalized size = 2.19 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c^{3} x^{5} + 1155 \, A b^{3} x + 385 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 1485 \,{\left (B b^{2} c + A b c^{2}\right )} x^{3} + 693 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^3*x^5 + 1155*A*b^3*x + 385*(3*B*b*c^2 + A*c^3)*x^4 + 1485*(B*b^2*c + A*b*c^2)*x^3 + 693*(B*b^3
+ 3*A*b^2*c)*x^2)*sqrt(x)

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Sympy [A]  time = 5.06267, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{3}{2}}}{3} + \frac{6 A b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 A b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A c^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} + \frac{6 B b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 B b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B c^{3} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**(5/2),x)

[Out]

2*A*b**3*x**(3/2)/3 + 6*A*b**2*c*x**(5/2)/5 + 6*A*b*c**2*x**(7/2)/7 + 2*A*c**3*x**(9/2)/9 + 2*B*b**3*x**(5/2)/
5 + 6*B*b**2*c*x**(7/2)/7 + 2*B*b*c**2*x**(9/2)/3 + 2*B*c**3*x**(11/2)/11

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Giac [A]  time = 1.12417, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{11} \, B c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b c^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, A b c^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{6}{5} \, A b^{2} c x^{\frac{5}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/x^(5/2),x, algorithm="giac")

[Out]

2/11*B*c^3*x^(11/2) + 2/3*B*b*c^2*x^(9/2) + 2/9*A*c^3*x^(9/2) + 6/7*B*b^2*c*x^(7/2) + 6/7*A*b*c^2*x^(7/2) + 2/
5*B*b^3*x^(5/2) + 6/5*A*b^2*c*x^(5/2) + 2/3*A*b^3*x^(3/2)