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

Optimal. Leaf size=59 \[ -\frac{2 A b^2}{3 x^{3/2}}-\frac{2 b (2 A c+b B)}{\sqrt{x}}+2 c \sqrt{x} (A c+2 b B)+\frac{2}{3} B c^2 x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0301225, antiderivative size = 59, 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} \[ -\frac{2 A b^2}{3 x^{3/2}}-\frac{2 b (2 A c+b B)}{\sqrt{x}}+2 c \sqrt{x} (A c+2 b B)+\frac{2}{3} B c^2 x^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0158453, size = 52, normalized size = 0.88 \[ \frac{2 B x \left (-3 b^2+6 b c x+c^2 x^2\right )-2 A \left (b^2+6 b c x-3 c^2 x^2\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 51, normalized size = 0.9 \begin{align*} -{\frac{-2\,B{c}^{2}{x}^{3}-6\,A{c}^{2}{x}^{2}-12\,B{x}^{2}bc+12\,Abcx+6\,{b}^{2}Bx+2\,A{b}^{2}}{3}{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)^2/x^(9/2),x)

[Out]

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

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Maxima [A]  time = 1.00288, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 2 \,{\left (2 \, B b c + A c^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A b^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{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)^2/x^(9/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.81532, size = 112, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (B c^{2} x^{3} - A b^{2} + 3 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{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)^2/x^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.53923, size = 73, normalized size = 1.24 \begin{align*} - \frac{2 A b^{2}}{3 x^{\frac{3}{2}}} - \frac{4 A b c}{\sqrt{x}} + 2 A c^{2} \sqrt{x} - \frac{2 B b^{2}}{\sqrt{x}} + 4 B b c \sqrt{x} + \frac{2 B c^{2} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11129, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{3} \, B c^{2} x^{\frac{3}{2}} + 4 \, B b c \sqrt{x} + 2 \, A c^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B b^{2} x + 6 \, A b c x + A b^{2}\right )}}{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)^2/x^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 4*B*b*c*sqrt(x) + 2*A*c^2*sqrt(x) - 2/3*(3*B*b^2*x + 6*A*b*c*x + A*b^2)/x^(3/2)