3.166 \(\int \frac{(A+B x) (b x+c x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=79 \[ 2 b^2 \sqrt{x} (3 A c+b B)-\frac{2 A b^3}{\sqrt{x}}+\frac{2}{5} c^2 x^{5/2} (A c+3 b B)+2 b c x^{3/2} (A c+b B)+\frac{2}{7} B c^3 x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0418045, antiderivative size = 79, 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} \[ 2 b^2 \sqrt{x} (3 A c+b B)-\frac{2 A b^3}{\sqrt{x}}+\frac{2}{5} c^2 x^{5/2} (A c+3 b B)+2 b c x^{3/2} (A c+b B)+\frac{2}{7} B c^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/x^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0210124, size = 75, normalized size = 0.95 \[ \frac{2 \left (7 A \left (15 b^2 c x-5 b^3+5 b c^2 x^2+c^3 x^3\right )+B x \left (35 b^2 c x+35 b^3+21 b c^2 x^2+5 c^3 x^3\right )\right )}{35 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/x^(9/2),x]

[Out]

(2*(7*A*(-5*b^3 + 15*b^2*c*x + 5*b*c^2*x^2 + c^3*x^3) + B*x*(35*b^3 + 35*b^2*c*x + 21*b*c^2*x^2 + 5*c^3*x^3)))
/(35*Sqrt[x])

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Maple [A]  time = 0.007, size = 76, normalized size = 1. \begin{align*} -{\frac{-10\,B{c}^{3}{x}^{4}-14\,A{x}^{3}{c}^{3}-42\,B{x}^{3}b{c}^{2}-70\,A{x}^{2}b{c}^{2}-70\,B{x}^{2}{b}^{2}c-210\,A{b}^{2}cx-70\,{b}^{3}Bx+70\,A{b}^{3}}{35}{\frac{1}{\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^(9/2),x)

[Out]

-2/35/x^(1/2)*(-5*B*c^3*x^4-7*A*c^3*x^3-21*B*b*c^2*x^3-35*A*b*c^2*x^2-35*B*b^2*c*x^2-105*A*b^2*c*x-35*B*b^3*x+
35*A*b^3)

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Maxima [A]  time = 1.01245, size = 99, normalized size = 1.25 \begin{align*} \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} - \frac{2 \, A b^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{5}{2}} + 2 \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{3} + 3 \, A b^{2} c\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^(9/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92262, size = 166, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (5 \, B c^{3} x^{4} - 35 \, A b^{3} + 7 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 35 \,{\left (B b^{2} c + A b c^{2}\right )} x^{2} + 35 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )}}{35 \, \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^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.45504, size = 105, normalized size = 1.33 \begin{align*} - \frac{2 A b^{3}}{\sqrt{x}} + 6 A b^{2} c \sqrt{x} + 2 A b c^{2} x^{\frac{3}{2}} + \frac{2 A c^{3} x^{\frac{5}{2}}}{5} + 2 B b^{3} \sqrt{x} + 2 B b^{2} c x^{\frac{3}{2}} + \frac{6 B b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B c^{3} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/x**(9/2),x)

[Out]

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

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Giac [A]  time = 1.1453, size = 104, normalized size = 1.32 \begin{align*} \frac{2}{7} \, B c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b c^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A c^{3} x^{\frac{5}{2}} + 2 \, B b^{2} c x^{\frac{3}{2}} + 2 \, A b c^{2} x^{\frac{3}{2}} + 2 \, B b^{3} \sqrt{x} + 6 \, A b^{2} c \sqrt{x} - \frac{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^(9/2),x, algorithm="giac")

[Out]

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