\(\int x^{5/2} (a+b x^2)^2 (c+d x^2)^3 \, dx\) [408]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 (2 b c+3 a d) x^{11/2}+\frac {2}{15} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+\frac {2}{19} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+\frac {2}{23} b d^2 (3 b c+2 a d) x^{23/2}+\frac {2}{27} b^2 d^3 x^{27/2} \]

[Out]

2/7*a^2*c^3*x^(7/2)+2/11*a*c^2*(3*a*d+2*b*c)*x^(11/2)+2/15*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*x^(15/2)+2/19*d*(a^
2*d^2+6*a*b*c*d+3*b^2*c^2)*x^(19/2)+2/23*b*d^2*(2*a*d+3*b*c)*x^(23/2)+2/27*b^2*d^3*x^(27/2)

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} \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{19} d x^{19/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{15} c x^{15/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac {2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac {2}{27} b^2 d^3 x^{27/2} \]

[In]

Int[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(7/2))/7 + (2*a*c^2*(2*b*c + 3*a*d)*x^(11/2))/11 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(15/2
))/15 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(19/2))/19 + (2*b*d^2*(3*b*c + 2*a*d)*x^(23/2))/23 + (2*b^2*d
^3*x^(27/2))/27

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 x^{5/2}+a c^2 (2 b c+3 a d) x^{9/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{13/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{17/2}+b d^2 (3 b c+2 a d) x^{21/2}+b^2 d^3 x^{25/2}\right ) \, dx \\ & = \frac {2}{7} a^2 c^3 x^{7/2}+\frac {2}{11} a c^2 (2 b c+3 a d) x^{11/2}+\frac {2}{15} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+\frac {2}{19} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+\frac {2}{23} b d^2 (3 b c+2 a d) x^{23/2}+\frac {2}{27} b^2 d^3 x^{27/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2 x^{7/2} \left (621 a^2 \left (1045 c^3+1995 c^2 d x^2+1463 c d^2 x^4+385 d^3 x^6\right )+378 a b x^2 \left (2185 c^3+4807 c^2 d x^2+3795 c d^2 x^4+1045 d^3 x^6\right )+77 b^2 x^4 \left (3933 c^3+9315 c^2 d x^2+7695 c d^2 x^4+2185 d^3 x^6\right )\right )}{4542615} \]

[In]

Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*x^(7/2)*(621*a^2*(1045*c^3 + 1995*c^2*d*x^2 + 1463*c*d^2*x^4 + 385*d^3*x^6) + 378*a*b*x^2*(2185*c^3 + 4807*
c^2*d*x^2 + 3795*c*d^2*x^4 + 1045*d^3*x^6) + 77*b^2*x^4*(3933*c^3 + 9315*c^2*d*x^2 + 7695*c*d^2*x^4 + 2185*d^3
*x^6)))/4542615

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7}\)	\(128\)
default	\(\frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {23}{2}}}{23}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7}\)	\(128\)
gosper	\(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 x^{6} d^{2} a b c +717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\)	\(138\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 x^{6} d^{2} a b c +717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\)	\(138\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (168245 b^{2} d^{3} x^{10}+395010 a b \,d^{3} x^{8}+592515 b^{2} c \,d^{2} x^{8}+239085 a^{2} d^{3} x^{6}+1434510 x^{6} d^{2} a b c +717255 b^{2} c^{2} d \,x^{6}+908523 a^{2} c \,d^{2} x^{4}+1817046 a b \,c^{2} d \,x^{4}+302841 b^{2} c^{3} x^{4}+1238895 a^{2} c^{2} d \,x^{2}+825930 a b \,c^{3} x^{2}+648945 a^{2} c^{3}\right )}{4542615}\)	\(138\)

[In]

int(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

2/27*b^2*d^3*x^(27/2)+2/23*(2*a*b*d^3+3*b^2*c*d^2)*x^(23/2)+2/19*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^(19/2)+2/
15*(3*a^2*c*d^2+6*a*b*c^2*d+b^2*c^3)*x^(15/2)+2/11*(3*a^2*c^2*d+2*a*b*c^3)*x^(11/2)+2/7*a^2*c^3*x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.95 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{4542615} \, {\left (168245 \, b^{2} d^{3} x^{13} + 197505 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + 239085 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + 648945 \, a^{2} c^{3} x^{3} + 302841 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + 412965 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5}\right )} \sqrt {x} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

2/4542615*(168245*b^2*d^3*x^13 + 197505*(3*b^2*c*d^2 + 2*a*b*d^3)*x^11 + 239085*(3*b^2*c^2*d + 6*a*b*c*d^2 + a
^2*d^3)*x^9 + 648945*a^2*c^3*x^3 + 302841*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^7 + 412965*(2*a*b*c^3 + 3*a^
2*c^2*d)*x^5)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.38 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2 a^{2} c^{3} x^{\frac {7}{2}}}{7} + \frac {6 a^{2} c^{2} d x^{\frac {11}{2}}}{11} + \frac {2 a^{2} c d^{2} x^{\frac {15}{2}}}{5} + \frac {2 a^{2} d^{3} x^{\frac {19}{2}}}{19} + \frac {4 a b c^{3} x^{\frac {11}{2}}}{11} + \frac {4 a b c^{2} d x^{\frac {15}{2}}}{5} + \frac {12 a b c d^{2} x^{\frac {19}{2}}}{19} + \frac {4 a b d^{3} x^{\frac {23}{2}}}{23} + \frac {2 b^{2} c^{3} x^{\frac {15}{2}}}{15} + \frac {6 b^{2} c^{2} d x^{\frac {19}{2}}}{19} + \frac {6 b^{2} c d^{2} x^{\frac {23}{2}}}{23} + \frac {2 b^{2} d^{3} x^{\frac {27}{2}}}{27} \]

[In]

integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

2*a**2*c**3*x**(7/2)/7 + 6*a**2*c**2*d*x**(11/2)/11 + 2*a**2*c*d**2*x**(15/2)/5 + 2*a**2*d**3*x**(19/2)/19 + 4
*a*b*c**3*x**(11/2)/11 + 4*a*b*c**2*d*x**(15/2)/5 + 12*a*b*c*d**2*x**(19/2)/19 + 4*a*b*d**3*x**(23/2)/23 + 2*b
**2*c**3*x**(15/2)/15 + 6*b**2*c**2*d*x**(19/2)/19 + 6*b**2*c*d**2*x**(23/2)/23 + 2*b**2*d**3*x**(27/2)/27

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{27} \, b^{2} d^{3} x^{\frac {27}{2}} + \frac {2}{23} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {19}{2}} + \frac {2}{7} \, a^{2} c^{3} x^{\frac {7}{2}} + \frac {2}{15} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {11}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

2/27*b^2*d^3*x^(27/2) + 2/23*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(23/2) + 2/19*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(19/2) + 2/7*a^2*c^3*x^(7/2) + 2/15*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(15/2) + 2/11*(2*a*b*c^3 + 3*a^2*
c^2*d)*x^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=\frac {2}{27} \, b^{2} d^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b^{2} c d^{2} x^{\frac {23}{2}} + \frac {4}{23} \, a b d^{3} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c^{2} d x^{\frac {19}{2}} + \frac {12}{19} \, a b c d^{2} x^{\frac {19}{2}} + \frac {2}{19} \, a^{2} d^{3} x^{\frac {19}{2}} + \frac {2}{15} \, b^{2} c^{3} x^{\frac {15}{2}} + \frac {4}{5} \, a b c^{2} d x^{\frac {15}{2}} + \frac {2}{5} \, a^{2} c d^{2} x^{\frac {15}{2}} + \frac {4}{11} \, a b c^{3} x^{\frac {11}{2}} + \frac {6}{11} \, a^{2} c^{2} d x^{\frac {11}{2}} + \frac {2}{7} \, a^{2} c^{3} x^{\frac {7}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")

[Out]

2/27*b^2*d^3*x^(27/2) + 6/23*b^2*c*d^2*x^(23/2) + 4/23*a*b*d^3*x^(23/2) + 6/19*b^2*c^2*d*x^(19/2) + 12/19*a*b*
c*d^2*x^(19/2) + 2/19*a^2*d^3*x^(19/2) + 2/15*b^2*c^3*x^(15/2) + 4/5*a*b*c^2*d*x^(15/2) + 2/5*a^2*c*d^2*x^(15/
2) + 4/11*a*b*c^3*x^(11/2) + 6/11*a^2*c^2*d*x^(11/2) + 2/7*a^2*c^3*x^(7/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx=x^{15/2}\,\left (\frac {2\,a^2\,c\,d^2}{5}+\frac {4\,a\,b\,c^2\,d}{5}+\frac {2\,b^2\,c^3}{15}\right )+x^{19/2}\,\left (\frac {2\,a^2\,d^3}{19}+\frac {12\,a\,b\,c\,d^2}{19}+\frac {6\,b^2\,c^2\,d}{19}\right )+\frac {2\,a^2\,c^3\,x^{7/2}}{7}+\frac {2\,b^2\,d^3\,x^{27/2}}{27}+\frac {2\,a\,c^2\,x^{11/2}\,\left (3\,a\,d+2\,b\,c\right )}{11}+\frac {2\,b\,d^2\,x^{23/2}\,\left (2\,a\,d+3\,b\,c\right )}{23} \]

[In]

int(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x)

[Out]

x^(15/2)*((2*b^2*c^3)/15 + (2*a^2*c*d^2)/5 + (4*a*b*c^2*d)/5) + x^(19/2)*((2*a^2*d^3)/19 + (6*b^2*c^2*d)/19 +
(12*a*b*c*d^2)/19) + (2*a^2*c^3*x^(7/2))/7 + (2*b^2*d^3*x^(27/2))/27 + (2*a*c^2*x^(11/2)*(3*a*d + 2*b*c))/11 +
(2*b*d^2*x^(23/2)*(2*a*d + 3*b*c))/23