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

Optimal. Leaf size=59 \[ -\frac{2 A b^2}{\sqrt{x}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt{x} (2 A c+b B)+\frac{2}{5} B c^2 x^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.029797, 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} \[ -\frac{2 A b^2}{\sqrt{x}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+2 b \sqrt{x} (2 A c+b B)+\frac{2}{5} B c^2 x^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0151144, size = 54, normalized size = 0.92 \[ \frac{10 A \left (-3 b^2+6 b c x+c^2 x^2\right )+2 B x \left (15 b^2+10 b c x+3 c^2 x^2\right )}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*A*(-3*b^2 + 6*b*c*x + c^2*x^2) + 2*B*x*(15*b^2 + 10*b*c*x + 3*c^2*x^2))/(15*Sqrt[x])

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Maple [A]  time = 0.004, size = 52, normalized size = 0.9 \begin{align*} -{\frac{-6\,B{c}^{2}{x}^{3}-10\,A{c}^{2}{x}^{2}-20\,B{x}^{2}bc-60\,Abcx-30\,{b}^{2}Bx+30\,A{b}^{2}}{15}{\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)^2/x^(7/2),x)

[Out]

-2/15/x^(1/2)*(-3*B*c^2*x^3-5*A*c^2*x^2-10*B*b*c*x^2-30*A*b*c*x-15*B*b^2*x+15*A*b^2)

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Maxima [A]  time = 0.998861, size = 69, normalized size = 1.17 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} - \frac{2 \, A b^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{2} + 2 \, A b 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)^2/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76893, size = 122, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{3} - 15 \, A b^{2} + 5 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x\right )}}{15 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c^2*x^3 - 15*A*b^2 + 5*(2*B*b*c + A*c^2)*x^2 + 15*(B*b^2 + 2*A*b*c)*x)/sqrt(x)

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Sympy [A]  time = 3.61825, size = 75, normalized size = 1.27 \begin{align*} - \frac{2 A b^{2}}{\sqrt{x}} + 4 A b c \sqrt{x} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} + 2 B b^{2} \sqrt{x} + \frac{4 B b c x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10995, size = 72, normalized size = 1.22 \begin{align*} \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B b c x^{\frac{3}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 2 \, B b^{2} \sqrt{x} + 4 \, A b c \sqrt{x} - \frac{2 \, A b^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/x^(7/2),x, algorithm="giac")

[Out]

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