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

Optimal. Leaf size=63 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(15/2))/1
5

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Rubi [A]  time = 0.0293865, antiderivative size = 63, 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} \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(15/2))/1
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 x^{3/2} (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{7/2}+b (b B+2 A c) x^{9/2}+c (2 b B+A c) x^{11/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac{2}{9} A b^2 x^{9/2}+\frac{2}{11} b (b B+2 A c) x^{11/2}+\frac{2}{13} c (2 b B+A c) x^{13/2}+\frac{2}{15} B c^2 x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0165733, size = 55, normalized size = 0.87 \[ \frac{2 x^{9/2} \left (5 A \left (143 b^2+234 b c x+99 c^2 x^2\right )+3 B x \left (195 b^2+330 b c x+143 c^2 x^2\right )\right )}{6435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(5*A*(143*b^2 + 234*b*c*x + 99*c^2*x^2) + 3*B*x*(195*b^2 + 330*b*c*x + 143*c^2*x^2)))/6435

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Maple [A]  time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{858\,B{c}^{2}{x}^{3}+990\,A{c}^{2}{x}^{2}+1980\,B{x}^{2}bc+2340\,Abcx+1170\,{b}^{2}Bx+1430\,A{b}^{2}}{6435}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6435*x^(9/2)*(429*B*c^2*x^3+495*A*c^2*x^2+990*B*b*c*x^2+1170*A*b*c*x+585*B*b^2*x+715*A*b^2)

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Maxima [A]  time = 1.09531, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*B*c^2*x^(15/2) + 2/9*A*b^2*x^(9/2) + 2/13*(2*B*b*c + A*c^2)*x^(13/2) + 2/11*(B*b^2 + 2*A*b*c)*x^(11/2)

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Fricas [A]  time = 1.80242, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{6435} \,{\left (429 \, B c^{2} x^{7} + 715 \, A b^{2} x^{4} + 495 \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + 585 \,{\left (B b^{2} + 2 \, A b c\right )} x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6435*(429*B*c^2*x^7 + 715*A*b^2*x^4 + 495*(2*B*b*c + A*c^2)*x^6 + 585*(B*b^2 + 2*A*b*c)*x^5)*sqrt(x)

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Sympy [A]  time = 4.71182, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b**2*x**(9/2)/9 + 4*A*b*c*x**(11/2)/11 + 2*A*c**2*x**(13/2)/13 + 2*B*b**2*x**(11/2)/11 + 4*B*b*c*x**(13/2)
/13 + 2*B*c**2*x**(15/2)/15

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Giac [A]  time = 1.10803, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B b c x^{\frac{13}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*B*c^2*x^(15/2) + 4/13*B*b*c*x^(13/2) + 2/13*A*c^2*x^(13/2) + 2/11*B*b^2*x^(11/2) + 4/11*A*b*c*x^(11/2) +
2/9*A*b^2*x^(9/2)