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

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

[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

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Rubi [A]  time = 0.0122232, antiderivative size = 45, normalized size of antiderivative = 1., 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} \[ \frac{2}{9} a A x^{9/2}+\frac{2}{11} a B x^{11/2}+\frac{2}{13} A c x^{13/2}+\frac{2}{15} B c x^{15/2} \]

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*}

Mathematica [A]  time = 0.0134621, size = 35, normalized size = 0.78 \[ \frac{2 x^{9/2} \left (65 a (11 A+9 B x)+33 c x^2 (15 A+13 B x)\right )}{6435} \]

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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Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{858\,Bc{x}^{3}+990\,Ac{x}^{2}+1170\,aBx+1430\,aA}{6435}{x}^{{\frac{9}{2}}}} \end{align*}

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 = 1.05694, size = 39, normalized size = 0.87 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.2583, size = 97, normalized size = 2.16 \begin{align*} \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{align*}

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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Sympy [A]  time = 9.95781, size = 46, normalized size = 1.02 \begin{align*} \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{align*}

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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Giac [A]  time = 1.29143, size = 39, normalized size = 0.87 \begin{align*} \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{align*}

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)