3.5.1 \(\int x^{3/2} (a+b x^2)^2 (c+d x^2)^2 \, dx\) [401]

Optimal. Leaf size=97 \[ \frac {2}{5} a^2 c^2 x^{5/2}+\frac {4}{9} a c (b c+a d) x^{9/2}+\frac {2}{13} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+\frac {4}{17} b d (b c+a d) x^{17/2}+\frac {2}{21} b^2 d^2 x^{21/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 = {459} \begin {gather*} \frac {2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{5} a^2 c^2 x^{5/2}+\frac {4}{17} b d x^{17/2} (a d+b c)+\frac {4}{9} a c x^{9/2} (a d+b c)+\frac {2}{21} b^2 d^2 x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 459

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 x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{3/2}+2 a c (b c+a d) x^{7/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+2 b d (b c+a d) x^{15/2}+b^2 d^2 x^{19/2}\right ) \, dx\\ &=\frac {2}{5} a^2 c^2 x^{5/2}+\frac {4}{9} a c (b c+a d) x^{9/2}+\frac {2}{13} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+\frac {4}{17} b d (b c+a d) x^{17/2}+\frac {2}{21} b^2 d^2 x^{21/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 93, normalized size = 0.96 \begin {gather*} \frac {2 x^{5/2} \left (119 a^2 \left (117 c^2+130 c d x^2+45 d^2 x^4\right )+70 a b x^2 \left (221 c^2+306 c d x^2+117 d^2 x^4\right )+15 b^2 x^4 \left (357 c^2+546 c d x^2+221 d^2 x^4\right )\right )}{69615} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(119*a^2*(117*c^2 + 130*c*d*x^2 + 45*d^2*x^4) + 70*a*b*x^2*(221*c^2 + 306*c*d*x^2 + 117*d^2*x^4) +
15*b^2*x^4*(357*c^2 + 546*c*d*x^2 + 221*d^2*x^4)))/69615

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 90, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {21}{2}}}{21}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c^{2} x^{\frac {5}{2}}}{5}\)	\(90\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {21}{2}}}{21}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} c^{2} x^{\frac {5}{2}}}{5}\)	\(90\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (3315 b^{2} d^{2} x^{8}+8190 a b \,d^{2} x^{6}+8190 b^{2} c d \,x^{6}+5355 a^{2} d^{2} x^{4}+21420 a b c d \,x^{4}+5355 b^{2} c^{2} x^{4}+15470 a^{2} c d \,x^{2}+15470 a b \,c^{2} x^{2}+13923 a^{2} c^{2}\right )}{69615}\)	\(97\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (3315 b^{2} d^{2} x^{8}+8190 a b \,d^{2} x^{6}+8190 b^{2} c d \,x^{6}+5355 a^{2} d^{2} x^{4}+21420 a b c d \,x^{4}+5355 b^{2} c^{2} x^{4}+15470 a^{2} c d \,x^{2}+15470 a b \,c^{2} x^{2}+13923 a^{2} c^{2}\right )}{69615}\)	\(97\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (3315 b^{2} d^{2} x^{8}+8190 a b \,d^{2} x^{6}+8190 b^{2} c d \,x^{6}+5355 a^{2} d^{2} x^{4}+21420 a b c d \,x^{4}+5355 b^{2} c^{2} x^{4}+15470 a^{2} c d \,x^{2}+15470 a b \,c^{2} x^{2}+13923 a^{2} c^{2}\right )}{69615}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 85, normalized size = 0.88 \begin {gather*} \frac {2}{21} \, b^{2} d^{2} x^{\frac {21}{2}} + \frac {4}{17} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} c^{2} x^{\frac {5}{2}} + \frac {4}{9} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.69, size = 90, normalized size = 0.93 \begin {gather*} \frac {2}{69615} \, {\left (3315 \, b^{2} d^{2} x^{10} + 8190 \, {\left (b^{2} c d + a b d^{2}\right )} x^{8} + 5355 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + 13923 \, a^{2} c^{2} x^{2} + 15470 \, {\left (a b c^{2} + a^{2} c d\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/69615*(3315*b^2*d^2*x^10 + 8190*(b^2*c*d + a*b*d^2)*x^8 + 5355*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^6 + 13923*a
^2*c^2*x^2 + 15470*(a*b*c^2 + a^2*c*d)*x^4)*sqrt(x)

________________________________________________________________________________________

Sympy [A]
time = 0.76, size = 136, normalized size = 1.40 \begin {gather*} \frac {2 a^{2} c^{2} x^{\frac {5}{2}}}{5} + \frac {4 a^{2} c d x^{\frac {9}{2}}}{9} + \frac {2 a^{2} d^{2} x^{\frac {13}{2}}}{13} + \frac {4 a b c^{2} x^{\frac {9}{2}}}{9} + \frac {8 a b c d x^{\frac {13}{2}}}{13} + \frac {4 a b d^{2} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c^{2} x^{\frac {13}{2}}}{13} + \frac {4 b^{2} c d x^{\frac {17}{2}}}{17} + \frac {2 b^{2} d^{2} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*c**2*x**(5/2)/5 + 4*a**2*c*d*x**(9/2)/9 + 2*a**2*d**2*x**(13/2)/13 + 4*a*b*c**2*x**(9/2)/9 + 8*a*b*c*d*
x**(13/2)/13 + 4*a*b*d**2*x**(17/2)/17 + 2*b**2*c**2*x**(13/2)/13 + 4*b**2*c*d*x**(17/2)/17 + 2*b**2*d**2*x**(
21/2)/21

________________________________________________________________________________________

Giac [A]
time = 0.95, size = 94, normalized size = 0.97 \begin {gather*} \frac {2}{21} \, b^{2} d^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b^{2} c d x^{\frac {17}{2}} + \frac {4}{17} \, a b d^{2} x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} c^{2} x^{\frac {13}{2}} + \frac {8}{13} \, a b c d x^{\frac {13}{2}} + \frac {2}{13} \, a^{2} d^{2} x^{\frac {13}{2}} + \frac {4}{9} \, a b c^{2} x^{\frac {9}{2}} + \frac {4}{9} \, a^{2} c d x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} c^{2} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b^2*d^2*x^(21/2) + 4/17*b^2*c*d*x^(17/2) + 4/17*a*b*d^2*x^(17/2) + 2/13*b^2*c^2*x^(13/2) + 8/13*a*b*c*d*x
^(13/2) + 2/13*a^2*d^2*x^(13/2) + 4/9*a*b*c^2*x^(9/2) + 4/9*a^2*c*d*x^(9/2) + 2/5*a^2*c^2*x^(5/2)

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 78, normalized size = 0.80 \begin {gather*} x^{13/2}\,\left (\frac {2\,a^2\,d^2}{13}+\frac {8\,a\,b\,c\,d}{13}+\frac {2\,b^2\,c^2}{13}\right )+\frac {2\,a^2\,c^2\,x^{5/2}}{5}+\frac {2\,b^2\,d^2\,x^{21/2}}{21}+\frac {4\,a\,c\,x^{9/2}\,\left (a\,d+b\,c\right )}{9}+\frac {4\,b\,d\,x^{17/2}\,\left (a\,d+b\,c\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(13/2)*((2*a^2*d^2)/13 + (2*b^2*c^2)/13 + (8*a*b*c*d)/13) + (2*a^2*c^2*x^(5/2))/5 + (2*b^2*d^2*x^(21/2))/21
+ (4*a*c*x^(9/2)*(a*d + b*c))/9 + (4*b*d*x^(17/2)*(a*d + b*c))/17

________________________________________________________________________________________