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

Optimal. Leaf size=137 \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]

[Out]

(-2*a^2*c^3)/Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(3/2))/3 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(7/2))/7
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(11/2))/11 + (2*b*d^2*(3*b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d^3*x^
(19/2))/19

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Rubi [A]  time = 0.0638812, 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} \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*c^3)/Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(3/2))/3 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(7/2))/7
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(11/2))/11 + (2*b*d^2*(3*b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d^3*x^
(19/2))/19

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^{3/2}} \, dx &=\int \left (\frac{a^2 c^3}{x^{3/2}}+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^{5/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{9/2}+b d^2 (3 b c+2 a d) x^{13/2}+b^2 d^3 x^{17/2}\right ) \, dx\\ &=-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 (2 b c+3 a d) x^{3/2}+\frac{2}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{7/2}+\frac{2}{11} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11/2}+\frac{2}{15} b d^2 (3 b c+2 a d) x^{15/2}+\frac{2}{19} b^2 d^3 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0513395, size = 137, normalized size = 1. \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*c^3)/Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(3/2))/3 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(7/2))/7
+ (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(11/2))/11 + (2*b*d^2*(3*b*c + 2*a*d)*x^(15/2))/15 + (2*b^2*d^3*x^
(19/2))/19

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Maple [A]  time = 0.007, size = 138, normalized size = 1. \begin{align*} -{\frac{-2310\,{b}^{2}{d}^{3}{x}^{10}-5852\,{x}^{8}ab{d}^{3}-8778\,{x}^{8}{b}^{2}c{d}^{2}-3990\,{x}^{6}{a}^{2}{d}^{3}-23940\,{x}^{6}abc{d}^{2}-11970\,{x}^{6}{b}^{2}{c}^{2}d-18810\,{x}^{4}{a}^{2}c{d}^{2}-37620\,{x}^{4}ab{c}^{2}d-6270\,{x}^{4}{b}^{2}{c}^{3}-43890\,{x}^{2}{a}^{2}{c}^{2}d-29260\,{x}^{2}ab{c}^{3}+43890\,{a}^{2}{c}^{3}}{21945}{\frac{1}{\sqrt{x}}}} \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^(3/2),x)

[Out]

-2/21945*(-1155*b^2*d^3*x^10-2926*a*b*d^3*x^8-4389*b^2*c*d^2*x^8-1995*a^2*d^3*x^6-11970*a*b*c*d^2*x^6-5985*b^2
*c^2*d*x^6-9405*a^2*c*d^2*x^4-18810*a*b*c^2*d*x^4-3135*b^2*c^3*x^4-21945*a^2*c^2*d*x^2-14630*a*b*c^3*x^2+21945
*a^2*c^3)/x^(1/2)

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Maxima [A]  time = 1.06686, size = 171, normalized size = 1.25 \begin{align*} \frac{2}{19} \, b^{2} d^{3} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{11}{2}} - \frac{2 \, a^{2} c^{3}}{\sqrt{x}} + \frac{2}{7} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} 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^(3/2),x, algorithm="maxima")

[Out]

2/19*b^2*d^3*x^(19/2) + 2/15*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(15/2) + 2/11*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(11/2) - 2*a^2*c^3/sqrt(x) + 2/7*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(7/2) + 2/3*(2*a*b*c^3 + 3*a^2*c^2*d
)*x^(3/2)

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Fricas [A]  time = 0.860847, size = 304, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (1155 \, b^{2} d^{3} x^{10} + 1463 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 1995 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 21945 \, a^{2} c^{3} + 3135 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 7315 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{21945 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/21945*(1155*b^2*d^3*x^10 + 1463*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 1995*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^6 - 21945*a^2*c^3 + 3135*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 7315*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/sqrt
(x)

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Sympy [A]  time = 11.564, size = 189, normalized size = 1.38 \begin{align*} - \frac{2 a^{2} c^{3}}{\sqrt{x}} + 2 a^{2} c^{2} d x^{\frac{3}{2}} + \frac{6 a^{2} c d^{2} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} d^{3} x^{\frac{11}{2}}}{11} + \frac{4 a b c^{3} x^{\frac{3}{2}}}{3} + \frac{12 a b c^{2} d x^{\frac{7}{2}}}{7} + \frac{12 a b c d^{2} x^{\frac{11}{2}}}{11} + \frac{4 a b d^{3} x^{\frac{15}{2}}}{15} + \frac{2 b^{2} c^{3} x^{\frac{7}{2}}}{7} + \frac{6 b^{2} c^{2} d x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c d^{2} x^{\frac{15}{2}}}{5} + \frac{2 b^{2} d^{3} x^{\frac{19}{2}}}{19} \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**(3/2),x)

[Out]

-2*a**2*c**3/sqrt(x) + 2*a**2*c**2*d*x**(3/2) + 6*a**2*c*d**2*x**(7/2)/7 + 2*a**2*d**3*x**(11/2)/11 + 4*a*b*c*
*3*x**(3/2)/3 + 12*a*b*c**2*d*x**(7/2)/7 + 12*a*b*c*d**2*x**(11/2)/11 + 4*a*b*d**3*x**(15/2)/15 + 2*b**2*c**3*
x**(7/2)/7 + 6*b**2*c**2*d*x**(11/2)/11 + 2*b**2*c*d**2*x**(15/2)/5 + 2*b**2*d**3*x**(19/2)/19

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Giac [A]  time = 1.15516, size = 182, normalized size = 1.33 \begin{align*} \frac{2}{19} \, b^{2} d^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c d^{2} x^{\frac{15}{2}} + \frac{4}{15} \, a b d^{3} x^{\frac{15}{2}} + \frac{6}{11} \, b^{2} c^{2} d x^{\frac{11}{2}} + \frac{12}{11} \, a b c d^{2} x^{\frac{11}{2}} + \frac{2}{11} \, a^{2} d^{3} x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c^{3} x^{\frac{7}{2}} + \frac{12}{7} \, a b c^{2} d x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c d^{2} x^{\frac{7}{2}} + \frac{4}{3} \, a b c^{3} x^{\frac{3}{2}} + 2 \, a^{2} c^{2} d x^{\frac{3}{2}} - \frac{2 \, a^{2} c^{3}}{\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^(3/2),x, algorithm="giac")

[Out]

2/19*b^2*d^3*x^(19/2) + 2/5*b^2*c*d^2*x^(15/2) + 4/15*a*b*d^3*x^(15/2) + 6/11*b^2*c^2*d*x^(11/2) + 12/11*a*b*c
*d^2*x^(11/2) + 2/11*a^2*d^3*x^(11/2) + 2/7*b^2*c^3*x^(7/2) + 12/7*a*b*c^2*d*x^(7/2) + 6/7*a^2*c*d^2*x^(7/2) +
4/3*a*b*c^3*x^(3/2) + 2*a^2*c^2*d*x^(3/2) - 2*a^2*c^3/sqrt(x)