\(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{x^{3/2}} \, dx\) [404]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=-\frac {2 a^2 c^2}{\sqrt {x}}+\frac {4}{3} a c (b c+a d) x^{3/2}+\frac {2}{7} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{7/2}+\frac {4}{11} b d (b c+a d) x^{11/2}+\frac {2}{15} b^2 d^2 x^{15/2} \]

[Out]

4/3*a*c*(a*d+b*c)*x^(3/2)+2/7*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(7/2)+4/11*b*d*(a*d+b*c)*x^(11/2)+2/15*b^2*d^2*x^(
15/2)-2*a^2*c^2/x^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 95, 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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=\frac {2}{7} x^{7/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {2 a^2 c^2}{\sqrt {x}}+\frac {4}{11} b d x^{11/2} (a d+b c)+\frac {4}{3} a c x^{3/2} (a d+b c)+\frac {2}{15} b^2 d^2 x^{15/2} \]

[In]

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

[Out]

(-2*a^2*c^2)/Sqrt[x] + (4*a*c*(b*c + a*d)*x^(3/2))/3 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(7/2))/7 + (4*b*d*
(b*c + a*d)*x^(11/2))/11 + (2*b^2*d^2*x^(15/2))/15

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=\frac {2 \left (-55 a^2 \left (21 c^2-14 c d x^2-3 d^2 x^4\right )+10 a b x^2 \left (77 c^2+66 c d x^2+21 d^2 x^4\right )+b^2 x^4 \left (165 c^2+210 c d x^2+77 d^2 x^4\right )\right )}{1155 \sqrt {x}} \]

[In]

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

[Out]

(2*(-55*a^2*(21*c^2 - 14*c*d*x^2 - 3*d^2*x^4) + 10*a*b*x^2*(77*c^2 + 66*c*d*x^2 + 21*d^2*x^4) + b^2*x^4*(165*c
^2 + 210*c*d*x^2 + 77*d^2*x^4)))/(1155*Sqrt[x])

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {15}{2}}}{15}+\frac {4 a b \,d^{2} x^{\frac {11}{2}}}{11}+\frac {4 b^{2} c d \,x^{\frac {11}{2}}}{11}+\frac {2 a^{2} d^{2} x^{\frac {7}{2}}}{7}+\frac {8 a b c d \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} c^{2} x^{\frac {7}{2}}}{7}+\frac {4 a^{2} c d \,x^{\frac {3}{2}}}{3}+\frac {4 a b \,c^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c^{2}}{\sqrt {x}}\)	\(95\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {15}{2}}}{15}+\frac {4 a b \,d^{2} x^{\frac {11}{2}}}{11}+\frac {4 b^{2} c d \,x^{\frac {11}{2}}}{11}+\frac {2 a^{2} d^{2} x^{\frac {7}{2}}}{7}+\frac {8 a b c d \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} c^{2} x^{\frac {7}{2}}}{7}+\frac {4 a^{2} c d \,x^{\frac {3}{2}}}{3}+\frac {4 a b \,c^{2} x^{\frac {3}{2}}}{3}-\frac {2 a^{2} c^{2}}{\sqrt {x}}\)	\(95\)
gosper	\(-\frac {2 \left (-77 b^{2} d^{2} x^{8}-210 a b \,d^{2} x^{6}-210 b^{2} c d \,x^{6}-165 a^{2} d^{2} x^{4}-660 x^{4} a b c d -165 b^{2} c^{2} x^{4}-770 a^{2} c d \,x^{2}-770 x^{2} b \,c^{2} a +1155 a^{2} c^{2}\right )}{1155 \sqrt {x}}\)	\(97\)
trager	\(-\frac {2 \left (-77 b^{2} d^{2} x^{8}-210 a b \,d^{2} x^{6}-210 b^{2} c d \,x^{6}-165 a^{2} d^{2} x^{4}-660 x^{4} a b c d -165 b^{2} c^{2} x^{4}-770 a^{2} c d \,x^{2}-770 x^{2} b \,c^{2} a +1155 a^{2} c^{2}\right )}{1155 \sqrt {x}}\)	\(97\)
risch	\(-\frac {2 \left (-77 b^{2} d^{2} x^{8}-210 a b \,d^{2} x^{6}-210 b^{2} c d \,x^{6}-165 a^{2} d^{2} x^{4}-660 x^{4} a b c d -165 b^{2} c^{2} x^{4}-770 a^{2} c d \,x^{2}-770 x^{2} b \,c^{2} a +1155 a^{2} c^{2}\right )}{1155 \sqrt {x}}\)	\(97\)

[In]

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

[Out]

2/15*b^2*d^2*x^(15/2)+4/11*a*b*d^2*x^(11/2)+4/11*b^2*c*d*x^(11/2)+2/7*a^2*d^2*x^(7/2)+8/7*a*b*c*d*x^(7/2)+2/7*
b^2*c^2*x^(7/2)+4/3*a^2*c*d*x^(3/2)+4/3*a*b*c^2*x^(3/2)-2*a^2*c^2/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=\frac {2 \, {\left (77 \, b^{2} d^{2} x^{8} + 210 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 165 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 1155 \, a^{2} c^{2} + 770 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, \sqrt {x}} \]

[In]

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

[Out]

2/1155*(77*b^2*d^2*x^8 + 210*(b^2*c*d + a*b*d^2)*x^6 + 165*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 1155*a^2*c^2
+ 770*(a*b*c^2 + a^2*c*d)*x^2)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=- \frac {2 a^{2} c^{2}}{\sqrt {x}} + \frac {4 a^{2} c d x^{\frac {3}{2}}}{3} + \frac {2 a^{2} d^{2} x^{\frac {7}{2}}}{7} + \frac {4 a b c^{2} x^{\frac {3}{2}}}{3} + \frac {8 a b c d x^{\frac {7}{2}}}{7} + \frac {4 a b d^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{2} c^{2} x^{\frac {7}{2}}}{7} + \frac {4 b^{2} c d x^{\frac {11}{2}}}{11} + \frac {2 b^{2} d^{2} x^{\frac {15}{2}}}{15} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=\frac {2}{15} \, b^{2} d^{2} x^{\frac {15}{2}} + \frac {4}{11} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {11}{2}} + \frac {2}{7} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, a^{2} c^{2}}{\sqrt {x}} + \frac {4}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {3}{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=\frac {2}{15} \, b^{2} d^{2} x^{\frac {15}{2}} + \frac {4}{11} \, b^{2} c d x^{\frac {11}{2}} + \frac {4}{11} \, a b d^{2} x^{\frac {11}{2}} + \frac {2}{7} \, b^{2} c^{2} x^{\frac {7}{2}} + \frac {8}{7} \, a b c d x^{\frac {7}{2}} + \frac {2}{7} \, a^{2} d^{2} x^{\frac {7}{2}} + \frac {4}{3} \, a b c^{2} x^{\frac {3}{2}} + \frac {4}{3} \, a^{2} c d x^{\frac {3}{2}} - \frac {2 \, a^{2} c^{2}}{\sqrt {x}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x^{3/2}} \, dx=x^{7/2}\,\left (\frac {2\,a^2\,d^2}{7}+\frac {8\,a\,b\,c\,d}{7}+\frac {2\,b^2\,c^2}{7}\right )-\frac {2\,a^2\,c^2}{\sqrt {x}}+\frac {2\,b^2\,d^2\,x^{15/2}}{15}+\frac {4\,a\,c\,x^{3/2}\,\left (a\,d+b\,c\right )}{3}+\frac {4\,b\,d\,x^{11/2}\,\left (a\,d+b\,c\right )}{11} \]

[In]

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

[Out]

x^(7/2)*((2*a^2*d^2)/7 + (2*b^2*c^2)/7 + (8*a*b*c*d)/7) - (2*a^2*c^2)/x^(1/2) + (2*b^2*d^2*x^(15/2))/15 + (4*a
*c*x^(3/2)*(a*d + b*c))/3 + (4*b*d*x^(11/2)*(a*d + b*c))/11