\(\int \sqrt {x} (a+b x^2)^2 (c+d x^2)^3 \, dx\) [410]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 (2 b c+3 a d) x^{7/2}+\frac {2}{11} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+\frac {2}{15} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+\frac {2}{19} b d^2 (3 b c+2 a d) x^{19/2}+\frac {2}{23} b^2 d^3 x^{23/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459} \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^3 x^{3/2}+\frac {2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac {2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac {2}{23} b^2 d^3 x^{23/2} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2 x^{3/2} \left (437 a^2 \left (385 c^3+495 c^2 d x^2+315 c d^2 x^4+77 d^3 x^6\right )+138 a b x^2 \left (1045 c^3+1995 c^2 d x^2+1463 c d^2 x^4+385 d^3 x^6\right )+21 b^2 x^4 \left (2185 c^3+4807 c^2 d x^2+3795 c d^2 x^4+1045 d^3 x^6\right )\right )}{504735} \]

[In]

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

[Out]

(2*x^(3/2)*(437*a^2*(385*c^3 + 495*c^2*d*x^2 + 315*c*d^2*x^4 + 77*d^3*x^6) + 138*a*b*x^2*(1045*c^3 + 1995*c^2*
d*x^2 + 1463*c*d^2*x^4 + 385*d^3*x^6) + 21*b^2*x^4*(2185*c^3 + 4807*c^2*d*x^2 + 3795*c*d^2*x^4 + 1045*d^3*x^6)
))/504735

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{3} x^{\frac {3}{2}}}{3}\)	\(128\)
default	\(\frac {2 b^{2} d^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{3} x^{\frac {3}{2}}}{3}\)	\(128\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (21945 b^{2} d^{3} x^{10}+53130 a b \,d^{3} x^{8}+79695 b^{2} c \,d^{2} x^{8}+33649 a^{2} d^{3} x^{6}+201894 x^{6} d^{2} a b c +100947 b^{2} c^{2} d \,x^{6}+137655 a^{2} c \,d^{2} x^{4}+275310 a b \,c^{2} d \,x^{4}+45885 b^{2} c^{3} x^{4}+216315 a^{2} c^{2} d \,x^{2}+144210 a b \,c^{3} x^{2}+168245 a^{2} c^{3}\right )}{504735}\)	\(138\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (21945 b^{2} d^{3} x^{10}+53130 a b \,d^{3} x^{8}+79695 b^{2} c \,d^{2} x^{8}+33649 a^{2} d^{3} x^{6}+201894 x^{6} d^{2} a b c +100947 b^{2} c^{2} d \,x^{6}+137655 a^{2} c \,d^{2} x^{4}+275310 a b \,c^{2} d \,x^{4}+45885 b^{2} c^{3} x^{4}+216315 a^{2} c^{2} d \,x^{2}+144210 a b \,c^{3} x^{2}+168245 a^{2} c^{3}\right )}{504735}\)	\(138\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (21945 b^{2} d^{3} x^{10}+53130 a b \,d^{3} x^{8}+79695 b^{2} c \,d^{2} x^{8}+33649 a^{2} d^{3} x^{6}+201894 x^{6} d^{2} a b c +100947 b^{2} c^{2} d \,x^{6}+137655 a^{2} c \,d^{2} x^{4}+275310 a b \,c^{2} d \,x^{4}+45885 b^{2} c^{3} x^{4}+216315 a^{2} c^{2} d \,x^{2}+144210 a b \,c^{3} x^{2}+168245 a^{2} c^{3}\right )}{504735}\)	\(138\)

[In]

int((b*x^2+a)^2*(d*x^2+c)^3*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{504735} \, {\left (21945 \, b^{2} d^{3} x^{11} + 26565 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + 33649 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + 168245 \, a^{2} c^{3} x + 45885 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + 72105 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\right )} \sqrt {x} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2 a^{2} c^{3} x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d^{3} x^{\frac {23}{2}}}{23} + \frac {2 x^{\frac {19}{2}} \cdot \left (2 a b d^{3} + 3 b^{2} c d^{2}\right )}{19} + \frac {2 x^{\frac {15}{2}} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{15} + \frac {2 x^{\frac {11}{2}} \cdot \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right )}{11} + \frac {2 x^{\frac {7}{2}} \cdot \left (3 a^{2} c^{2} d + 2 a b c^{3}\right )}{7} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{23} \, b^{2} d^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {15}{2}} + \frac {2}{3} \, a^{2} c^{3} x^{\frac {3}{2}} + \frac {2}{11} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {7}{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{23} \, b^{2} d^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c d^{2} x^{\frac {19}{2}} + \frac {4}{19} \, a b d^{3} x^{\frac {19}{2}} + \frac {2}{5} \, b^{2} c^{2} d x^{\frac {15}{2}} + \frac {4}{5} \, a b c d^{2} x^{\frac {15}{2}} + \frac {2}{15} \, a^{2} d^{3} x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c^{3} x^{\frac {11}{2}} + \frac {12}{11} \, a b c^{2} d x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b c^{3} x^{\frac {7}{2}} + \frac {6}{7} \, a^{2} c^{2} d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c^{3} x^{\frac {3}{2}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^{11/2}\,\left (\frac {6\,a^2\,c\,d^2}{11}+\frac {12\,a\,b\,c^2\,d}{11}+\frac {2\,b^2\,c^3}{11}\right )+x^{15/2}\,\left (\frac {2\,a^2\,d^3}{15}+\frac {4\,a\,b\,c\,d^2}{5}+\frac {2\,b^2\,c^2\,d}{5}\right )+\frac {2\,a^2\,c^3\,x^{3/2}}{3}+\frac {2\,b^2\,d^3\,x^{23/2}}{23}+\frac {2\,a\,c^2\,x^{7/2}\,\left (3\,a\,d+2\,b\,c\right )}{7}+\frac {2\,b\,d^2\,x^{19/2}\,\left (2\,a\,d+3\,b\,c\right )}{19} \]

[In]

int(x^(1/2)*(a + b*x^2)^2*(c + d*x^2)^3,x)

[Out]

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