3.405 \(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac {2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{3 x^{3/2}}+\frac {4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt {x} (a d+b c)+\frac {2}{13} b^2 d^2 x^{13/2} \]

[Out]

-2/3*a^2*c^2/x^(3/2)+2/5*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(5/2)+4/9*b*d*(a*d+b*c)*x^(9/2)+2/13*b^2*d^2*x^(13/2)+4
*a*c*(a*d+b*c)*x^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {448} \[ \frac {2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{3 x^{3/2}}+\frac {4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt {x} (a d+b c)+\frac {2}{13} b^2 d^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]

[Out]

(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2))/5 + (4*b*d*
(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*x^(13/2))/13

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 \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac {a^2 c^2}{x^{5/2}}+\frac {2 a c (b c+a d)}{\sqrt {x}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+2 b d (b c+a d) x^{7/2}+b^2 d^2 x^{11/2}\right ) \, dx\\ &=-\frac {2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt {x}+\frac {2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac {4}{9} b d (b c+a d) x^{9/2}+\frac {2}{13} b^2 d^2 x^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 83, normalized size = 0.87 \[ \frac {2 \left (117 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-195 a^2 c^2+130 b d x^6 (a d+b c)+1170 a c x^2 (a d+b c)+45 b^2 d^2 x^8\right )}{585 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]

[Out]

(2*(-195*a^2*c^2 + 1170*a*c*(b*c + a*d)*x^2 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 130*b*d*(b*c + a*d)*x^
6 + 45*b^2*d^2*x^8))/(585*x^(3/2))

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fricas [A]  time = 0.42, size = 87, normalized size = 0.92 \[ \frac {2 \, {\left (45 \, b^{2} d^{2} x^{8} + 130 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d^2*x^8 + 130*(b^2*c*d + a*b*d^2)*x^6 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 195*a^2*c^2 +
1170*(a*b*c^2 + a^2*c*d)*x^2)/x^(3/2)

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giac [A]  time = 0.37, size = 94, normalized size = 0.99 \[ \frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b^{2} c d x^{\frac {9}{2}} + \frac {4}{9} \, a b d^{2} x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c^{2} x^{\frac {5}{2}} + \frac {8}{5} \, a b c d x^{\frac {5}{2}} + \frac {2}{5} \, a^{2} d^{2} x^{\frac {5}{2}} + 4 \, a b c^{2} \sqrt {x} + 4 \, a^{2} c d \sqrt {x} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="giac")

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*b^2*c*d*x^(9/2) + 4/9*a*b*d^2*x^(9/2) + 2/5*b^2*c^2*x^(5/2) + 8/5*a*b*c*d*x^(5/2)
+ 2/5*a^2*d^2*x^(5/2) + 4*a*b*c^2*sqrt(x) + 4*a^2*c*d*sqrt(x) - 2/3*a^2*c^2/x^(3/2)

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maple [A]  time = 0.01, size = 97, normalized size = 1.02 \[ -\frac {2 \left (-45 b^{2} d^{2} x^{8}-130 a b \,d^{2} x^{6}-130 b^{2} c d \,x^{6}-117 a^{2} d^{2} x^{4}-468 a b c d \,x^{4}-117 b^{2} c^{2} x^{4}-1170 a^{2} c d \,x^{2}-1170 a b \,c^{2} x^{2}+195 a^{2} c^{2}\right )}{585 x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x)

[Out]

-2/585*(-45*b^2*d^2*x^8-130*a*b*d^2*x^6-130*b^2*c*d*x^6-117*a^2*d^2*x^4-468*a*b*c*d*x^4-117*b^2*c^2*x^4-1170*a
^2*c*d*x^2-1170*a*b*c^2*x^2+195*a^2*c^2)/x^(3/2)

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maxima [A]  time = 1.01, size = 85, normalized size = 0.89 \[ \frac {2}{13} \, b^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, a^{2} c^{2}}{3 \, x^{\frac {3}{2}}} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="maxima")

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*(b^2*c*d + a*b*d^2)*x^(9/2) + 2/5*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2) - 2/3*a^
2*c^2/x^(3/2) + 4*(a*b*c^2 + a^2*c*d)*sqrt(x)

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mupad [B]  time = 0.03, size = 78, normalized size = 0.82 \[ x^{5/2}\,\left (\frac {2\,a^2\,d^2}{5}+\frac {8\,a\,b\,c\,d}{5}+\frac {2\,b^2\,c^2}{5}\right )-\frac {2\,a^2\,c^2}{3\,x^{3/2}}+\frac {2\,b^2\,d^2\,x^{13/2}}{13}+4\,a\,c\,\sqrt {x}\,\left (a\,d+b\,c\right )+\frac {4\,b\,d\,x^{9/2}\,\left (a\,d+b\,c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x)

[Out]

x^(5/2)*((2*a^2*d^2)/5 + (2*b^2*c^2)/5 + (8*a*b*c*d)/5) - (2*a^2*c^2)/(3*x^(3/2)) + (2*b^2*d^2*x^(13/2))/13 +
4*a*c*x^(1/2)*(a*d + b*c) + (4*b*d*x^(9/2)*(a*d + b*c))/9

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sympy [A]  time = 6.07, size = 133, normalized size = 1.40 \[ - \frac {2 a^{2} c^{2}}{3 x^{\frac {3}{2}}} + 4 a^{2} c d \sqrt {x} + \frac {2 a^{2} d^{2} x^{\frac {5}{2}}}{5} + 4 a b c^{2} \sqrt {x} + \frac {8 a b c d x^{\frac {5}{2}}}{5} + \frac {4 a b d^{2} x^{\frac {9}{2}}}{9} + \frac {2 b^{2} c^{2} x^{\frac {5}{2}}}{5} + \frac {4 b^{2} c d x^{\frac {9}{2}}}{9} + \frac {2 b^{2} d^{2} x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(5/2),x)

[Out]

-2*a**2*c**2/(3*x**(3/2)) + 4*a**2*c*d*sqrt(x) + 2*a**2*d**2*x**(5/2)/5 + 4*a*b*c**2*sqrt(x) + 8*a*b*c*d*x**(5
/2)/5 + 4*a*b*d**2*x**(9/2)/9 + 2*b**2*c**2*x**(5/2)/5 + 4*b**2*c*d*x**(9/2)/9 + 2*b**2*d**2*x**(13/2)/13

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