3.391 \(\int x^{7/2} (a+b x^2)^2 (c+d x^2) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} b x^{17/2} (2 a d+b c)+\frac{2}{13} a x^{13/2} (a d+2 b c)+\frac{2}{21} b^2 d x^{21/2} \]

[Out]

(2*a^2*c*x^(9/2))/9 + (2*a*(2*b*c + a*d)*x^(13/2))/13 + (2*b*(b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d*x^(21/2))/2
1

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Rubi [A]  time = 0.0327075, 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 = {448} \[ \frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} b x^{17/2} (2 a d+b c)+\frac{2}{13} a x^{13/2} (a d+2 b c)+\frac{2}{21} b^2 d x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(2*a^2*c*x^(9/2))/9 + (2*a*(2*b*c + a*d)*x^(13/2))/13 + (2*b*(b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d*x^(21/2))/2
1

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^{7/2}+a (2 b c+a d) x^{11/2}+b (b c+2 a d) x^{15/2}+b^2 d x^{19/2}\right ) \, dx\\ &=\frac{2}{9} a^2 c x^{9/2}+\frac{2}{13} a (2 b c+a d) x^{13/2}+\frac{2}{17} b (b c+2 a d) x^{17/2}+\frac{2}{21} b^2 d x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0324698, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 a^2 c+819 b x^4 (2 a d+b c)+1071 a x^2 (a d+2 b c)+663 b^2 d x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(2*x^(9/2)*(1547*a^2*c + 1071*a*(2*b*c + a*d)*x^2 + 819*b*(b*c + 2*a*d)*x^4 + 663*b^2*d*x^6))/13923

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{1326\,{b}^{2}d{x}^{6}+3276\,{x}^{4}abd+1638\,{b}^{2}c{x}^{4}+2142\,{x}^{2}{a}^{2}d+4284\,abc{x}^{2}+3094\,{a}^{2}c}{13923}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x)

[Out]

2/13923*x^(9/2)*(663*b^2*d*x^6+1638*a*b*d*x^4+819*b^2*c*x^4+1071*a^2*d*x^2+2142*a*b*c*x^2+1547*a^2*c)

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Maxima [A]  time = 1.04874, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{21} \, b^{2} d x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{17}{2}} + \frac{2}{9} \, a^{2} c x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b^2*d*x^(21/2) + 2/17*(b^2*c + 2*a*b*d)*x^(17/2) + 2/9*a^2*c*x^(9/2) + 2/13*(2*a*b*c + a^2*d)*x^(13/2)

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Fricas [A]  time = 0.755708, size = 146, normalized size = 2.32 \begin{align*} \frac{2}{13923} \,{\left (663 \, b^{2} d x^{10} + 819 \,{\left (b^{2} c + 2 \, a b d\right )} x^{8} + 1547 \, a^{2} c x^{4} + 1071 \,{\left (2 \, a b c + a^{2} d\right )} x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13923*(663*b^2*d*x^10 + 819*(b^2*c + 2*a*b*d)*x^8 + 1547*a^2*c*x^4 + 1071*(2*a*b*c + a^2*d)*x^6)*sqrt(x)

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Sympy [A]  time = 20.147, size = 80, normalized size = 1.27 \begin{align*} \frac{2 a^{2} c x^{\frac{9}{2}}}{9} + \frac{2 a^{2} d x^{\frac{13}{2}}}{13} + \frac{4 a b c x^{\frac{13}{2}}}{13} + \frac{4 a b d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

2*a**2*c*x**(9/2)/9 + 2*a**2*d*x**(13/2)/13 + 4*a*b*c*x**(13/2)/13 + 4*a*b*d*x**(17/2)/17 + 2*b**2*c*x**(17/2)
/17 + 2*b**2*d*x**(21/2)/21

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Giac [A]  time = 1.15688, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{21} \, b^{2} d x^{\frac{21}{2}} + \frac{2}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{4}{17} \, a b d x^{\frac{17}{2}} + \frac{4}{13} \, a b c x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, a^{2} c x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b^2*d*x^(21/2) + 2/17*b^2*c*x^(17/2) + 4/17*a*b*d*x^(17/2) + 4/13*a*b*c*x^(13/2) + 2/13*a^2*d*x^(13/2) +
2/9*a^2*c*x^(9/2)