3.413 \(\int \frac{(a+b x^2)^2 (c+d x^2)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt{x} (3 a d+2 b c)+\frac{2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac{2}{17} b^2 d^3 x^{17/2} \]

[Out]

(-2*a^2*c^3)/(3*x^(3/2)) + 2*a*c^2*(2*b*c + 3*a*d)*Sqrt[x] + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2))/5
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (2*b*d^2*(3*b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d^3*x^(1
7/2))/17

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Rubi [A]  time = 0.0644083, antiderivative size = 137, normalized size of antiderivative = 1., 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} \[ \frac{2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt{x} (3 a d+2 b c)+\frac{2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac{2}{17} b^2 d^3 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]

[Out]

(-2*a^2*c^3)/(3*x^(3/2)) + 2*a*c^2*(2*b*c + 3*a*d)*Sqrt[x] + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2))/5
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (2*b*d^2*(3*b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d^3*x^(1
7/2))/17

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 )^3}{x^{5/2}} \, dx &=\int \left (\frac{a^2 c^3}{x^{5/2}}+\frac{a c^2 (2 b c+3 a d)}{\sqrt{x}}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{3/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{7/2}+b d^2 (3 b c+2 a d) x^{11/2}+b^2 d^3 x^{15/2}\right ) \, dx\\ &=-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 (2 b c+3 a d) \sqrt{x}+\frac{2}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{5/2}+\frac{2}{9} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{9/2}+\frac{2}{13} b d^2 (3 b c+2 a d) x^{13/2}+\frac{2}{17} b^2 d^3 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0523587, size = 137, normalized size = 1. \[ \frac{2}{9} d x^{9/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{5} c x^{5/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{3 x^{3/2}}+2 a c^2 \sqrt{x} (3 a d+2 b c)+\frac{2}{13} b d^2 x^{13/2} (2 a d+3 b c)+\frac{2}{17} b^2 d^3 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(5/2),x]

[Out]

(-2*a^2*c^3)/(3*x^(3/2)) + 2*a*c^2*(2*b*c + 3*a*d)*Sqrt[x] + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(5/2))/5
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (2*b*d^2*(3*b*c + 2*a*d)*x^(13/2))/13 + (2*b^2*d^3*x^(1
7/2))/17

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Maple [A]  time = 0.007, size = 138, normalized size = 1. \begin{align*} -{\frac{-1170\,{b}^{2}{d}^{3}{x}^{10}-3060\,{x}^{8}ab{d}^{3}-4590\,{x}^{8}{b}^{2}c{d}^{2}-2210\,{x}^{6}{a}^{2}{d}^{3}-13260\,{x}^{6}abc{d}^{2}-6630\,{x}^{6}{b}^{2}{c}^{2}d-11934\,{x}^{4}{a}^{2}c{d}^{2}-23868\,{x}^{4}ab{c}^{2}d-3978\,{x}^{4}{b}^{2}{c}^{3}-59670\,{x}^{2}{a}^{2}{c}^{2}d-39780\,{x}^{2}ab{c}^{3}+6630\,{a}^{2}{c}^{3}}{9945}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x)

[Out]

-2/9945*(-585*b^2*d^3*x^10-1530*a*b*d^3*x^8-2295*b^2*c*d^2*x^8-1105*a^2*d^3*x^6-6630*a*b*c*d^2*x^6-3315*b^2*c^
2*d*x^6-5967*a^2*c*d^2*x^4-11934*a*b*c^2*d*x^4-1989*b^2*c^3*x^4-29835*a^2*c^2*d*x^2-19890*a*b*c^3*x^2+3315*a^2
*c^3)/x^(3/2)

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Maxima [A]  time = 1.08122, size = 171, normalized size = 1.25 \begin{align*} \frac{2}{17} \, b^{2} d^{3} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{9}{2}} - \frac{2 \, a^{2} c^{3}}{3 \, x^{\frac{3}{2}}} + \frac{2}{5} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{5}{2}} + 2 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/17*b^2*d^3*x^(17/2) + 2/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(13/2) + 2/9*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^
(9/2) - 2/3*a^2*c^3/x^(3/2) + 2/5*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(5/2) + 2*(2*a*b*c^3 + 3*a^2*c^2*d)*
sqrt(x)

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Fricas [A]  time = 0.905874, size = 298, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (585 \, b^{2} d^{3} x^{10} + 765 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 1105 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 3315 \, a^{2} c^{3} + 1989 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 9945 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{9945 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/9945*(585*b^2*d^3*x^10 + 765*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 1105*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6
- 3315*a^2*c^3 + 1989*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 9945*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x^(3/2)

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Sympy [A]  time = 12.5521, size = 189, normalized size = 1.38 \begin{align*} - \frac{2 a^{2} c^{3}}{3 x^{\frac{3}{2}}} + 6 a^{2} c^{2} d \sqrt{x} + \frac{6 a^{2} c d^{2} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d^{3} x^{\frac{9}{2}}}{9} + 4 a b c^{3} \sqrt{x} + \frac{12 a b c^{2} d x^{\frac{5}{2}}}{5} + \frac{4 a b c d^{2} x^{\frac{9}{2}}}{3} + \frac{4 a b d^{3} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c^{3} x^{\frac{5}{2}}}{5} + \frac{2 b^{2} c^{2} d x^{\frac{9}{2}}}{3} + \frac{6 b^{2} c d^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d^{3} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**3/x**(5/2),x)

[Out]

-2*a**2*c**3/(3*x**(3/2)) + 6*a**2*c**2*d*sqrt(x) + 6*a**2*c*d**2*x**(5/2)/5 + 2*a**2*d**3*x**(9/2)/9 + 4*a*b*
c**3*sqrt(x) + 12*a*b*c**2*d*x**(5/2)/5 + 4*a*b*c*d**2*x**(9/2)/3 + 4*a*b*d**3*x**(13/2)/13 + 2*b**2*c**3*x**(
5/2)/5 + 2*b**2*c**2*d*x**(9/2)/3 + 6*b**2*c*d**2*x**(13/2)/13 + 2*b**2*d**3*x**(17/2)/17

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Giac [A]  time = 1.18935, size = 182, normalized size = 1.33 \begin{align*} \frac{2}{17} \, b^{2} d^{3} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c d^{2} x^{\frac{13}{2}} + \frac{4}{13} \, a b d^{3} x^{\frac{13}{2}} + \frac{2}{3} \, b^{2} c^{2} d x^{\frac{9}{2}} + \frac{4}{3} \, a b c d^{2} x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d^{3} x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} c^{3} x^{\frac{5}{2}} + \frac{12}{5} \, a b c^{2} d x^{\frac{5}{2}} + \frac{6}{5} \, a^{2} c d^{2} x^{\frac{5}{2}} + 4 \, a b c^{3} \sqrt{x} + 6 \, a^{2} c^{2} d \sqrt{x} - \frac{2 \, a^{2} c^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(5/2),x, algorithm="giac")

[Out]

2/17*b^2*d^3*x^(17/2) + 6/13*b^2*c*d^2*x^(13/2) + 4/13*a*b*d^3*x^(13/2) + 2/3*b^2*c^2*d*x^(9/2) + 4/3*a*b*c*d^
2*x^(9/2) + 2/9*a^2*d^3*x^(9/2) + 2/5*b^2*c^3*x^(5/2) + 12/5*a*b*c^2*d*x^(5/2) + 6/5*a^2*c*d^2*x^(5/2) + 4*a*b
*c^3*sqrt(x) + 6*a^2*c^2*d*sqrt(x) - 2/3*a^2*c^3/x^(3/2)