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

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

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Rubi [A]  time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 55, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.04, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (715 A b^2 x^{9/2}+1170 A b c x^{11/2}+495 A c^2 x^{13/2}+585 b^2 B x^{11/2}+990 b B c x^{13/2}+429 B c^2 x^{15/2}\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(715*A*b^2*x^(9/2) + 585*b^2*B*x^(11/2) + 1170*A*b*c*x^(11/2) + 990*b*B*c*x^(13/2) + 495*A*c^2*x^(13/2) + 4
29*B*c^2*x^(15/2)))/6435

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fricas [A]  time = 0.39, size = 56, normalized size = 0.89 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 53, normalized size = 0.84 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 \left (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}\right ) x^{\frac {9}{2}}}{6435} \end {gather*}

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 = 0.54, size = 51, normalized size = 0.81 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,b^2}{11}+\frac {4\,A\,c\,b}{11}\right )+x^{13/2}\,\left (\frac {2\,A\,c^2}{13}+\frac {4\,B\,b\,c}{13}\right )+\frac {2\,A\,b^2\,x^{9/2}}{9}+\frac {2\,B\,c^2\,x^{15/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.49, size = 80, normalized size = 1.27 \begin {gather*} \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 {gather*}

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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