3.4.84 \(\int x^{7/2} (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=45 \[ \frac {2}{9} a A x^{9/2}+\frac {2}{11} a B x^{11/2}+\frac {2}{13} A c x^{13/2}+\frac {2}{15} B c x^{15/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} \frac {2}{9} a A x^{9/2}+\frac {2}{11} a B x^{11/2}+\frac {2}{13} A c x^{13/2}+\frac {2}{15} B c x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x)*(a + c*x^2),x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.78 \begin {gather*} \frac {2 x^{9/2} \left (65 a (11 A+9 B x)+33 c x^2 (15 A+13 B x)\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(65*a*(11*A + 9*B*x) + 33*c*x^2*(15*A + 13*B*x)))/6435

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \frac {2 \left (715 a A x^{9/2}+585 a B x^{11/2}+495 A c x^{13/2}+429 B c x^{15/2}\right )}{6435} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(A + B*x)*(a + c*x^2),x]

[Out]

(2*(715*a*A*x^(9/2) + 585*a*B*x^(11/2) + 495*A*c*x^(13/2) + 429*B*c*x^(15/2)))/6435

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fricas [A]  time = 0.41, size = 34, normalized size = 0.76 \begin {gather*} \frac {2}{6435} \, {\left (429 \, B c x^{7} + 495 \, A c x^{6} + 585 \, B a x^{5} + 715 \, A a x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.23, size = 29, normalized size = 0.64 \begin {gather*} \frac {2}{15} \, B c x^{\frac {15}{2}} + \frac {2}{13} \, A c x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 30, normalized size = 0.67 \begin {gather*} \frac {2 \left (429 B c \,x^{3}+495 A c \,x^{2}+585 B a x +715 a A \right ) x^{\frac {9}{2}}}{6435} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)*(c*x^2+a),x)

[Out]

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

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maxima [A]  time = 0.46, size = 29, normalized size = 0.64 \begin {gather*} \frac {2}{15} \, B c x^{\frac {15}{2}} + \frac {2}{13} \, A c x^{\frac {13}{2}} + \frac {2}{11} \, B a x^{\frac {11}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 29, normalized size = 0.64 \begin {gather*} \frac {2\,A\,a\,x^{9/2}}{9}+\frac {2\,B\,a\,x^{11/2}}{11}+\frac {2\,A\,c\,x^{13/2}}{13}+\frac {2\,B\,c\,x^{15/2}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + c*x^2)*(A + B*x),x)

[Out]

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

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sympy [A]  time = 8.22, size = 46, normalized size = 1.02 \begin {gather*} \frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A c x^{\frac {13}{2}}}{13} + \frac {2 B a x^{\frac {11}{2}}}{11} + \frac {2 B c x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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