3.414 \(\int \frac{(a+b x^2)^2 (c+d x^2)^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{2}{7} d x^{7/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{3} c x^{3/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{5 x^{5/2}}-\frac{2 a c^2 (3 a d+2 b c)}{\sqrt{x}}+\frac{2}{11} b d^2 x^{11/2} (2 a d+3 b c)+\frac{2}{15} b^2 d^3 x^{15/2} \]

[Out]

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

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Rubi [A]  time = 0.0627778, 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} \[ \frac{2}{7} d x^{7/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{3} c x^{3/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{5 x^{5/2}}-\frac{2 a c^2 (3 a d+2 b c)}{\sqrt{x}}+\frac{2}{11} b d^2 x^{11/2} (2 a d+3 b c)+\frac{2}{15} b^2 d^3 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^(7/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0481395, size = 121, normalized size = 0.88 \[ \frac{2 \left (165 d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+385 c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-231 a^2 c^3-1155 a c^2 x^2 (3 a d+2 b c)+105 b d^2 x^8 (2 a d+3 b c)+77 b^2 d^3 x^{10}\right )}{1155 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(7/2),x]

[Out]

(2*(-231*a^2*c^3 - 1155*a*c^2*(2*b*c + 3*a*d)*x^2 + 385*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^4 + 165*d*(3*b^2
*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 105*b*d^2*(3*b*c + 2*a*d)*x^8 + 77*b^2*d^3*x^10))/(1155*x^(5/2))

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Maple [A]  time = 0.006, size = 138, normalized size = 1. \begin{align*} -{\frac{-154\,{b}^{2}{d}^{3}{x}^{10}-420\,{x}^{8}ab{d}^{3}-630\,{x}^{8}{b}^{2}c{d}^{2}-330\,{x}^{6}{a}^{2}{d}^{3}-1980\,{x}^{6}abc{d}^{2}-990\,{x}^{6}{b}^{2}{c}^{2}d-2310\,{x}^{4}{a}^{2}c{d}^{2}-4620\,{x}^{4}ab{c}^{2}d-770\,{x}^{4}{b}^{2}{c}^{3}+6930\,{x}^{2}{a}^{2}{c}^{2}d+4620\,{x}^{2}ab{c}^{3}+462\,{a}^{2}{c}^{3}}{1155}{x}^{-{\frac{5}{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^(7/2),x)

[Out]

-2/1155*(-77*b^2*d^3*x^10-210*a*b*d^3*x^8-315*b^2*c*d^2*x^8-165*a^2*d^3*x^6-990*a*b*c*d^2*x^6-495*b^2*c^2*d*x^
6-1155*a^2*c*d^2*x^4-2310*a*b*c^2*d*x^4-385*b^2*c^3*x^4+3465*a^2*c^2*d*x^2+2310*a*b*c^3*x^2+231*a^2*c^3)/x^(5/
2)

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Maxima [A]  time = 1.06531, size = 174, normalized size = 1.27 \begin{align*} \frac{2}{15} \, b^{2} d^{3} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c^{3} + 5 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.806183, size = 293, normalized size = 2.14 \begin{align*} \frac{2 \,{\left (77 \, b^{2} d^{3} x^{10} + 105 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 165 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 231 \, a^{2} c^{3} + 385 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 1155 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{1155 \, x^{\frac{5}{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^(7/2),x, algorithm="fricas")

[Out]

2/1155*(77*b^2*d^3*x^10 + 105*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 165*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6 -
231*a^2*c^3 + 385*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 - 1155*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x^(5/2)

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Sympy [A]  time = 16.2194, size = 185, normalized size = 1.35 \begin{align*} - \frac{2 a^{2} c^{3}}{5 x^{\frac{5}{2}}} - \frac{6 a^{2} c^{2} d}{\sqrt{x}} + 2 a^{2} c d^{2} x^{\frac{3}{2}} + \frac{2 a^{2} d^{3} x^{\frac{7}{2}}}{7} - \frac{4 a b c^{3}}{\sqrt{x}} + 4 a b c^{2} d x^{\frac{3}{2}} + \frac{12 a b c d^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b d^{3} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c^{3} x^{\frac{3}{2}}}{3} + \frac{6 b^{2} c^{2} d x^{\frac{7}{2}}}{7} + \frac{6 b^{2} c d^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} d^{3} x^{\frac{15}{2}}}{15} \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**(7/2),x)

[Out]

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

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Giac [A]  time = 1.16136, size = 185, normalized size = 1.35 \begin{align*} \frac{2}{15} \, b^{2} d^{3} x^{\frac{15}{2}} + \frac{6}{11} \, b^{2} c d^{2} x^{\frac{11}{2}} + \frac{4}{11} \, a b d^{3} x^{\frac{11}{2}} + \frac{6}{7} \, b^{2} c^{2} d x^{\frac{7}{2}} + \frac{12}{7} \, a b c d^{2} x^{\frac{7}{2}} + \frac{2}{7} \, a^{2} d^{3} x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c^{3} x^{\frac{3}{2}} + 4 \, a b c^{2} d x^{\frac{3}{2}} + 2 \, a^{2} c d^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c^{3} x^{2} + 15 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="giac")

[Out]

2/15*b^2*d^3*x^(15/2) + 6/11*b^2*c*d^2*x^(11/2) + 4/11*a*b*d^3*x^(11/2) + 6/7*b^2*c^2*d*x^(7/2) + 12/7*a*b*c*d
^2*x^(7/2) + 2/7*a^2*d^3*x^(7/2) + 2/3*b^2*c^3*x^(3/2) + 4*a*b*c^2*d*x^(3/2) + 2*a^2*c*d^2*x^(3/2) - 2/5*(10*a
*b*c^3*x^2 + 15*a^2*c^2*d*x^2 + a^2*c^3)/x^(5/2)