\(\int \sqrt {x} (a+b x^2)^2 (c+d x^2)^2 \, dx\) [402]

Optimal result

Integrand size = 24, antiderivative size = 97 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{7} a c (b c+a d) x^{7/2}+\frac {2}{11} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+\frac {4}{15} b d (b c+a d) x^{15/2}+\frac {2}{19} b^2 d^2 x^{19/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{3} a^2 c^2 x^{3/2}+\frac {4}{15} b d x^{15/2} (a d+b c)+\frac {4}{7} a c x^{7/2} (a d+b c)+\frac {2}{19} b^2 d^2 x^{19/2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 x^{3/2} \left (95 a^2 \left (77 c^2+66 c d x^2+21 d^2 x^4\right )+38 a b x^2 \left (165 c^2+210 c d x^2+77 d^2 x^4\right )+7 b^2 x^4 \left (285 c^2+418 c d x^2+165 d^2 x^4\right )\right )}{21945} \]

[In]

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

[Out]

(2*x^(3/2)*(95*a^2*(77*c^2 + 66*c*d*x^2 + 21*d^2*x^4) + 38*a*b*x^2*(165*c^2 + 210*c*d*x^2 + 77*d^2*x^4) + 7*b^
2*x^4*(285*c^2 + 418*c*d*x^2 + 165*d^2*x^4)))/21945

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3}\)	\(90\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3}\)	\(90\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 x^{4} a b c d +1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 x^{2} b \,c^{2} a +7315 a^{2} c^{2}\right )}{21945}\)	\(97\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 x^{4} a b c d +1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 x^{2} b \,c^{2} a +7315 a^{2} c^{2}\right )}{21945}\)	\(97\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (1155 b^{2} d^{2} x^{8}+2926 a b \,d^{2} x^{6}+2926 b^{2} c d \,x^{6}+1995 a^{2} d^{2} x^{4}+7980 x^{4} a b c d +1995 b^{2} c^{2} x^{4}+6270 a^{2} c d \,x^{2}+6270 x^{2} b \,c^{2} a +7315 a^{2} c^{2}\right )}{21945}\)	\(97\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{21945} \, {\left (1155 \, b^{2} d^{2} x^{9} + 2926 \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + 1995 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + 7315 \, a^{2} c^{2} x + 6270 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2*x^(1/2),x, algorithm="fricas")

[Out]

2/21945*(1155*b^2*d^2*x^9 + 2926*(b^2*c*d + a*b*d^2)*x^7 + 1995*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5 + 7315*a^2
*c^2*x + 6270*(a*b*c^2 + a^2*c*d)*x^3)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 a^{2} c^{2} x^{\frac {3}{2}}}{3} + \frac {2 b^{2} d^{2} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {15}{2}} \cdot \left (2 a b d^{2} + 2 b^{2} c d\right )}{15} + \frac {2 x^{\frac {11}{2}} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{11} + \frac {2 x^{\frac {7}{2}} \cdot \left (2 a^{2} c d + 2 a b c^{2}\right )}{7} \]

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**2*x**(1/2),x)

[Out]

2*a**2*c**2*x**(3/2)/3 + 2*b**2*d**2*x**(19/2)/19 + 2*x**(15/2)*(2*a*b*d**2 + 2*b**2*c*d)/15 + 2*x**(11/2)*(a*
*2*d**2 + 4*a*b*c*d + b**2*c**2)/11 + 2*x**(7/2)*(2*a**2*c*d + 2*a*b*c**2)/7

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} + \frac {4}{7} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {7}{2}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2*x^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{19} \, b^{2} d^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b^{2} c d x^{\frac {15}{2}} + \frac {4}{15} \, a b d^{2} x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} c^{2} x^{\frac {11}{2}} + \frac {8}{11} \, a b c d x^{\frac {11}{2}} + \frac {2}{11} \, a^{2} d^{2} x^{\frac {11}{2}} + \frac {4}{7} \, a b c^{2} x^{\frac {7}{2}} + \frac {4}{7} \, a^{2} c d x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} c^{2} x^{\frac {3}{2}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^2*x^(1/2),x, algorithm="giac")

[Out]

2/19*b^2*d^2*x^(19/2) + 4/15*b^2*c*d*x^(15/2) + 4/15*a*b*d^2*x^(15/2) + 2/11*b^2*c^2*x^(11/2) + 8/11*a*b*c*d*x
^(11/2) + 2/11*a^2*d^2*x^(11/2) + 4/7*a*b*c^2*x^(7/2) + 4/7*a^2*c*d*x^(7/2) + 2/3*a^2*c^2*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int \sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^{11/2}\,\left (\frac {2\,a^2\,d^2}{11}+\frac {8\,a\,b\,c\,d}{11}+\frac {2\,b^2\,c^2}{11}\right )+\frac {2\,a^2\,c^2\,x^{3/2}}{3}+\frac {2\,b^2\,d^2\,x^{19/2}}{19}+\frac {4\,a\,c\,x^{7/2}\,\left (a\,d+b\,c\right )}{7}+\frac {4\,b\,d\,x^{15/2}\,\left (a\,d+b\,c\right )}{15} \]

[In]

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

[Out]

x^(11/2)*((2*a^2*d^2)/11 + (2*b^2*c^2)/11 + (8*a*b*c*d)/11) + (2*a^2*c^2*x^(3/2))/3 + (2*b^2*d^2*x^(19/2))/19
+ (4*a*c*x^(7/2)*(a*d + b*c))/7 + (4*b*d*x^(15/2)*(a*d + b*c))/15