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

Optimal. Leaf size=137 \[ \frac{2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt{x}+\frac{2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac{2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac{2}{21} b^2 d^3 x^{21/2} \]

[Out]

2*a^2*c^3*Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(5/2))/5 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2))/9 +
(2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (2*b*d^2*(3*b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d^3*x^(21
/2))/21

________________________________________________________________________________________

Rubi [A]  time = 0.0647974, 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} \[ \frac{2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt{x}+\frac{2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac{2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac{2}{21} b^2 d^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*c^3*Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(5/2))/5 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2))/9 +
(2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (2*b*d^2*(3*b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d^3*x^(21
/2))/21

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

Mathematica [A]  time = 0.0388549, size = 137, normalized size = 1. \[ \frac{2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt{x}+\frac{2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac{2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac{2}{21} b^2 d^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*c^3*Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(5/2))/5 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2))/9 +
(2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (2*b*d^2*(3*b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d^3*x^(21
/2))/21

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 138, normalized size = 1. \begin{align*}{\frac{6630\,{b}^{2}{d}^{3}{x}^{10}+16380\,{x}^{8}ab{d}^{3}+24570\,{x}^{8}{b}^{2}c{d}^{2}+10710\,{x}^{6}{a}^{2}{d}^{3}+64260\,{x}^{6}abc{d}^{2}+32130\,{x}^{6}{b}^{2}{c}^{2}d+46410\,{x}^{4}{a}^{2}c{d}^{2}+92820\,{x}^{4}ab{c}^{2}d+15470\,{x}^{4}{b}^{2}{c}^{3}+83538\,{x}^{2}{a}^{2}{c}^{2}d+55692\,{x}^{2}ab{c}^{3}+139230\,{a}^{2}{c}^{3}}{69615}\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^(1/2),x)

[Out]

2/69615*x^(1/2)*(3315*b^2*d^3*x^10+8190*a*b*d^3*x^8+12285*b^2*c*d^2*x^8+5355*a^2*d^3*x^6+32130*a*b*c*d^2*x^6+1
6065*b^2*c^2*d*x^6+23205*a^2*c*d^2*x^4+46410*a*b*c^2*d*x^4+7735*b^2*c^3*x^4+41769*a^2*c^2*d*x^2+27846*a*b*c^3*
x^2+69615*a^2*c^3)

________________________________________________________________________________________

Maxima [A]  time = 1.089, size = 171, normalized size = 1.25 \begin{align*} \frac{2}{21} \, b^{2} d^{3} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{13}{2}} + 2 \, a^{2} c^{3} \sqrt{x} + \frac{2}{9} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} 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^(1/2),x, algorithm="maxima")

[Out]

2/21*b^2*d^3*x^(21/2) + 2/17*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(17/2) + 2/13*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(13/2) + 2*a^2*c^3*sqrt(x) + 2/9*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(9/2) + 2/5*(2*a*b*c^3 + 3*a^2*c^2*d
)*x^(5/2)

________________________________________________________________________________________

Fricas [A]  time = 0.821759, size = 305, normalized size = 2.23 \begin{align*} \frac{2}{69615} \,{\left (3315 \, b^{2} d^{3} x^{10} + 4095 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 5355 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + 69615 \, a^{2} c^{3} + 7735 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 13923 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\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]

2/69615*(3315*b^2*d^3*x^10 + 4095*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 5355*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^6 + 69615*a^2*c^3 + 7735*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 13923*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)*sqr
t(x)

________________________________________________________________________________________

Sympy [A]  time = 10.4738, size = 190, normalized size = 1.39 \begin{align*} 2 a^{2} c^{3} \sqrt{x} + \frac{6 a^{2} c^{2} d x^{\frac{5}{2}}}{5} + \frac{2 a^{2} c d^{2} x^{\frac{9}{2}}}{3} + \frac{2 a^{2} d^{3} x^{\frac{13}{2}}}{13} + \frac{4 a b c^{3} x^{\frac{5}{2}}}{5} + \frac{4 a b c^{2} d x^{\frac{9}{2}}}{3} + \frac{12 a b c d^{2} x^{\frac{13}{2}}}{13} + \frac{4 a b d^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c^{2} d x^{\frac{13}{2}}}{13} + \frac{6 b^{2} c d^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d^{3} x^{\frac{21}{2}}}{21} \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*sqrt(x) + 6*a**2*c**2*d*x**(5/2)/5 + 2*a**2*c*d**2*x**(9/2)/3 + 2*a**2*d**3*x**(13/2)/13 + 4*a*b*c
**3*x**(5/2)/5 + 4*a*b*c**2*d*x**(9/2)/3 + 12*a*b*c*d**2*x**(13/2)/13 + 4*a*b*d**3*x**(17/2)/17 + 2*b**2*c**3*
x**(9/2)/9 + 6*b**2*c**2*d*x**(13/2)/13 + 6*b**2*c*d**2*x**(17/2)/17 + 2*b**2*d**3*x**(21/2)/21

________________________________________________________________________________________

Giac [A]  time = 1.69149, size = 182, normalized size = 1.33 \begin{align*} \frac{2}{21} \, b^{2} d^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c d^{2} x^{\frac{17}{2}} + \frac{4}{17} \, a b d^{3} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c^{2} d x^{\frac{13}{2}} + \frac{12}{13} \, a b c d^{2} x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d^{3} x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c^{3} x^{\frac{9}{2}} + \frac{4}{3} \, a b c^{2} d x^{\frac{9}{2}} + \frac{2}{3} \, a^{2} c d^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b c^{3} x^{\frac{5}{2}} + \frac{6}{5} \, a^{2} c^{2} d x^{\frac{5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/21*b^2*d^3*x^(21/2) + 6/17*b^2*c*d^2*x^(17/2) + 4/17*a*b*d^3*x^(17/2) + 6/13*b^2*c^2*d*x^(13/2) + 12/13*a*b*
c*d^2*x^(13/2) + 2/13*a^2*d^3*x^(13/2) + 2/9*b^2*c^3*x^(9/2) + 4/3*a*b*c^2*d*x^(9/2) + 2/3*a^2*c*d^2*x^(9/2) +
4/5*a*b*c^3*x^(5/2) + 6/5*a^2*c^2*d*x^(5/2) + 2*a^2*c^3*sqrt(x)