3.410 \(\int \sqrt{x} (a+b x^2)^2 (c+d x^2)^3 \, dx\)

Optimal. Leaf size=139 \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]

[Out]

(2*a^2*c^3*x^(3/2))/3 + (2*a*c^2*(2*b*c + 3*a*d)*x^(7/2))/7 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(11/2))
/11 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (2*b*d^2*(3*b*c + 2*a*d)*x^(19/2))/19 + (2*b^2*d^3
*x^(23/2))/23

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Rubi [A]  time = 0.0635255, antiderivative size = 139, 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} \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*c^3*x^(3/2))/3 + (2*a*c^2*(2*b*c + 3*a*d)*x^(7/2))/7 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(11/2))
/11 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (2*b*d^2*(3*b*c + 2*a*d)*x^(19/2))/19 + (2*b^2*d^3
*x^(23/2))/23

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 \sqrt{x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 \sqrt{x}+a c^2 (2 b c+3 a d) x^{5/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+b d^2 (3 b c+2 a d) x^{17/2}+b^2 d^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 (2 b c+3 a d) x^{7/2}+\frac{2}{11} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+\frac{2}{15} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+\frac{2}{19} b d^2 (3 b c+2 a d) x^{19/2}+\frac{2}{23} b^2 d^3 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.0375953, size = 139, normalized size = 1. \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*c^3*x^(3/2))/3 + (2*a*c^2*(2*b*c + 3*a*d)*x^(7/2))/7 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(11/2))
/11 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (2*b*d^2*(3*b*c + 2*a*d)*x^(19/2))/19 + (2*b^2*d^3
*x^(23/2))/23

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Maple [A]  time = 0.007, size = 138, normalized size = 1. \begin{align*}{\frac{43890\,{b}^{2}{d}^{3}{x}^{10}+106260\,{x}^{8}ab{d}^{3}+159390\,{x}^{8}{b}^{2}c{d}^{2}+67298\,{x}^{6}{a}^{2}{d}^{3}+403788\,{x}^{6}abc{d}^{2}+201894\,{x}^{6}{b}^{2}{c}^{2}d+275310\,{x}^{4}{a}^{2}c{d}^{2}+550620\,{x}^{4}ab{c}^{2}d+91770\,{x}^{4}{b}^{2}{c}^{3}+432630\,{x}^{2}{a}^{2}{c}^{2}d+288420\,{x}^{2}ab{c}^{3}+336490\,{a}^{2}{c}^{3}}{504735}{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^(1/2),x)

[Out]

2/504735*x^(3/2)*(21945*b^2*d^3*x^10+53130*a*b*d^3*x^8+79695*b^2*c*d^2*x^8+33649*a^2*d^3*x^6+201894*a*b*c*d^2*
x^6+100947*b^2*c^2*d*x^6+137655*a^2*c*d^2*x^4+275310*a*b*c^2*d*x^4+45885*b^2*c^3*x^4+216315*a^2*c^2*d*x^2+1442
10*a*b*c^3*x^2+168245*a^2*c^3)

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Maxima [A]  time = 1.04112, size = 171, normalized size = 1.23 \begin{align*} \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{15}{2}} + \frac{2}{3} \, a^{2} c^{3} x^{\frac{3}{2}} + \frac{2}{11} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{7}{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^(1/2),x, algorithm="maxima")

[Out]

2/23*b^2*d^3*x^(23/2) + 2/19*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(19/2) + 2/15*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(15/2) + 2/3*a^2*c^3*x^(3/2) + 2/11*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(11/2) + 2/7*(2*a*b*c^3 + 3*a^2*c
^2*d)*x^(7/2)

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Fricas [A]  time = 0.80791, size = 316, normalized size = 2.27 \begin{align*} \frac{2}{504735} \,{\left (21945 \, b^{2} d^{3} x^{11} + 26565 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + 33649 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + 168245 \, a^{2} c^{3} x + 45885 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + 72105 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\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^(1/2),x, algorithm="fricas")

[Out]

2/504735*(21945*b^2*d^3*x^11 + 26565*(3*b^2*c*d^2 + 2*a*b*d^3)*x^9 + 33649*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^
3)*x^7 + 168245*a^2*c^3*x + 45885*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^5 + 72105*(2*a*b*c^3 + 3*a^2*c^2*d)*
x^3)*sqrt(x)

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Sympy [A]  time = 5.20731, size = 155, normalized size = 1.12 \begin{align*} \frac{2 a^{2} c^{3} x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d^{3} x^{\frac{23}{2}}}{23} + \frac{2 x^{\frac{19}{2}} \left (2 a b d^{3} + 3 b^{2} c d^{2}\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (3 a^{2} c^{2} d + 2 a b c^{3}\right )}{7} \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**(1/2),x)

[Out]

2*a**2*c**3*x**(3/2)/3 + 2*b**2*d**3*x**(23/2)/23 + 2*x**(19/2)*(2*a*b*d**3 + 3*b**2*c*d**2)/19 + 2*x**(15/2)*
(a**2*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d)/15 + 2*x**(11/2)*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3)/11 + 2*
x**(7/2)*(3*a**2*c**2*d + 2*a*b*c**3)/7

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Giac [A]  time = 1.17829, size = 182, normalized size = 1.31 \begin{align*} \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c d^{2} x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c^{2} d x^{\frac{15}{2}} + \frac{4}{5} \, a b c d^{2} x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{3} x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c^{3} x^{\frac{11}{2}} + \frac{12}{11} \, a b c^{2} d x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b c^{3} x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c^{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^(1/2),x, algorithm="giac")

[Out]

2/23*b^2*d^3*x^(23/2) + 6/19*b^2*c*d^2*x^(19/2) + 4/19*a*b*d^3*x^(19/2) + 2/5*b^2*c^2*d*x^(15/2) + 4/5*a*b*c*d
^2*x^(15/2) + 2/15*a^2*d^3*x^(15/2) + 2/11*b^2*c^3*x^(11/2) + 12/11*a*b*c^2*d*x^(11/2) + 6/11*a^2*c*d^2*x^(11/
2) + 4/7*a*b*c^3*x^(7/2) + 6/7*a^2*c^2*d*x^(7/2) + 2/3*a^2*c^3*x^(3/2)