3.165 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{2}{3} b^2 x^{3/2} (3 A c+b B)+2 A b^3 \sqrt{x}+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{6}{5} b c x^{5/2} (A c+b B)+\frac{2}{9} B c^3 x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.043835, antiderivative size = 83, 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} \[ \frac{2}{3} b^2 x^{3/2} (3 A c+b B)+2 A b^3 \sqrt{x}+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{6}{5} b c x^{5/2} (A c+b B)+\frac{2}{9} B c^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0453847, size = 63, normalized size = 0.76 \[ \frac{2 \sqrt{x} \left (35 B (b+c x)^4-\left (35 b^2 c x+35 b^3+21 b c^2 x^2+5 c^3 x^3\right ) (b B-9 A c)\right )}{315 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(35*B*(b + c*x)^4 - (b*B - 9*A*c)*(35*b^3 + 35*b^2*c*x + 21*b*c^2*x^2 + 5*c^3*x^3)))/(315*c)

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Maple [A]  time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{70\,B{c}^{3}{x}^{4}+90\,A{x}^{3}{c}^{3}+270\,B{x}^{3}b{c}^{2}+378\,A{x}^{2}b{c}^{2}+378\,B{x}^{2}{b}^{2}c+630\,A{b}^{2}cx+210\,{b}^{3}Bx+630\,A{b}^{3}}{315}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/x^(7/2),x)

[Out]

2/315*x^(1/2)*(35*B*c^3*x^4+45*A*c^3*x^3+135*B*b*c^2*x^3+189*A*b*c^2*x^2+189*B*b^2*c*x^2+315*A*b^2*c*x+105*B*b
^3*x+315*A*b^3)

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Maxima [A]  time = 1.10509, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + 2 \, A b^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} 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^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.7079, size = 174, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \, B c^{3} x^{4} + 315 \, A b^{3} + 45 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 189 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 105 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x\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^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^3*x^4 + 315*A*b^3 + 45*(3*B*b*c^2 + A*c^3)*x^3 + 189*(B*b^2*c + A*b*c^2)*x^2 + 105*(B*b^3 + 3*A*
b^2*c)*x)*sqrt(x)

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Sympy [A]  time = 6.90168, size = 110, normalized size = 1.33 \begin{align*} 2 A b^{3} \sqrt{x} + 2 A b^{2} c x^{\frac{3}{2}} + \frac{6 A b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A c^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{3}{2}}}{3} + \frac{6 B b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**(7/2),x)

[Out]

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

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Giac [A]  time = 1.10846, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b c^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b^{2} c x^{\frac{5}{2}} + \frac{6}{5} \, A b c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 2 \, A b^{2} c x^{\frac{3}{2}} + 2 \, A b^{3} \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^(7/2),x, algorithm="giac")

[Out]

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