\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{\sqrt {x}} \, dx\) [411]

Optimal result

Integrand size = 24, antiderivative size = 137 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 (2 b c+3 a d) x^{5/2}+\frac {2}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+\frac {2}{13} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+\frac {2}{17} b d^2 (3 b c+2 a d) x^{17/2}+\frac {2}{21} b^2 d^3 x^{21/2} \]

[Out]

2/5*a*c^2*(3*a*d+2*b*c)*x^(5/2)+2/9*c*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*x^(9/2)+2/13*d*(a^2*d^2+6*a*b*c*d+3*b^2*c^
2)*x^(13/2)+2/17*b*d^2*(2*a*d+3*b*c)*x^(17/2)+2/21*b^2*d^3*x^(21/2)+2*a^2*c^3*x^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 137, 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} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac {2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac {2}{21} b^2 d^3 x^{21/2} \]

[In]

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

[Out]

2*a^2*c^3*Sqrt[x] + (2*a*c^2*(2*b*c + 3*a*d)*x^(5/2))/5 + (2*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^(9/2))/9 +
(2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (2*b*d^2*(3*b*c + 2*a*d)*x^(17/2))/17 + (2*b^2*d^3*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*} \text {integral}& = \int \left (\frac {a^2 c^3}{\sqrt {x}}+a c^2 (2 b c+3 a d) x^{3/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{7/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11/2}+b d^2 (3 b c+2 a d) x^{15/2}+b^2 d^3 x^{19/2}\right ) \, dx \\ & = 2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 (2 b c+3 a d) x^{5/2}+\frac {2}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+\frac {2}{13} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+\frac {2}{17} b d^2 (3 b c+2 a d) x^{17/2}+\frac {2}{21} b^2 d^3 x^{21/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (357 a^2 \left (195 c^3+117 c^2 d x^2+65 c d^2 x^4+15 d^3 x^6\right )+42 a b x^2 \left (663 c^3+1105 c^2 d x^2+765 c d^2 x^4+195 d^3 x^6\right )+5 b^2 x^4 \left (1547 c^3+3213 c^2 d x^2+2457 c d^2 x^4+663 d^3 x^6\right )\right )}{69615} \]

[In]

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

[Out]

(2*Sqrt[x]*(357*a^2*(195*c^3 + 117*c^2*d*x^2 + 65*c*d^2*x^4 + 15*d^3*x^6) + 42*a*b*x^2*(663*c^3 + 1105*c^2*d*x
^2 + 765*c*d^2*x^4 + 195*d^3*x^6) + 5*b^2*x^4*(1547*c^3 + 3213*c^2*d*x^2 + 2457*c*d^2*x^4 + 663*d^3*x^6)))/696
15

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{3} x^{\frac {21}{2}}}{21}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {5}{2}}}{5}+2 a^{2} c^{3} \sqrt {x}\)	\(128\)
default	\(\frac {2 b^{2} d^{3} x^{\frac {21}{2}}}{21}+\frac {2 \left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{\frac {5}{2}}}{5}+2 a^{2} c^{3} \sqrt {x}\)	\(128\)
trager	\(\left (\frac {2}{21} b^{2} d^{3} x^{10}+\frac {4}{17} a b \,d^{3} x^{8}+\frac {6}{17} b^{2} c \,d^{2} x^{8}+\frac {2}{13} a^{2} d^{3} x^{6}+\frac {12}{13} x^{6} d^{2} a b c +\frac {6}{13} b^{2} c^{2} d \,x^{6}+\frac {2}{3} a^{2} c \,d^{2} x^{4}+\frac {4}{3} a b \,c^{2} d \,x^{4}+\frac {2}{9} b^{2} c^{3} x^{4}+\frac {6}{5} a^{2} c^{2} d \,x^{2}+\frac {4}{5} a b \,c^{3} x^{2}+2 a^{2} c^{3}\right ) \sqrt {x}\)	\(137\)
gosper	\(\frac {2 \sqrt {x}\, \left (3315 b^{2} d^{3} x^{10}+8190 a b \,d^{3} x^{8}+12285 b^{2} c \,d^{2} x^{8}+5355 a^{2} d^{3} x^{6}+32130 x^{6} d^{2} a b c +16065 b^{2} c^{2} d \,x^{6}+23205 a^{2} c \,d^{2} x^{4}+46410 a b \,c^{2} d \,x^{4}+7735 b^{2} c^{3} x^{4}+41769 a^{2} c^{2} d \,x^{2}+27846 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right )}{69615}\)	\(138\)
risch	\(\frac {2 \sqrt {x}\, \left (3315 b^{2} d^{3} x^{10}+8190 a b \,d^{3} x^{8}+12285 b^{2} c \,d^{2} x^{8}+5355 a^{2} d^{3} x^{6}+32130 x^{6} d^{2} a b c +16065 b^{2} c^{2} d \,x^{6}+23205 a^{2} c \,d^{2} x^{4}+46410 a b \,c^{2} d \,x^{4}+7735 b^{2} c^{3} x^{4}+41769 a^{2} c^{2} d \,x^{2}+27846 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right )}{69615}\)	\(138\)

[In]

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

[Out]

2/21*b^2*d^3*x^(21/2)+2/17*(2*a*b*d^3+3*b^2*c*d^2)*x^(17/2)+2/13*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^(13/2)+2/
9*(3*a^2*c*d^2+6*a*b*c^2*d+b^2*c^3)*x^(9/2)+2/5*(3*a^2*c^2*d+2*a*b*c^3)*x^(5/2)+2*a^2*c^3*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{69615} \, {\left (3315 \, b^{2} d^{3} x^{10} + 4095 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 5355 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + 69615 \, a^{2} c^{3} + 7735 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 13923 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/69615*(3315*b^2*d^3*x^10 + 4095*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 5355*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^6 + 69615*a^2*c^3 + 7735*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 13923*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)*sqr
t(x)

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=2 a^{2} c^{3} \sqrt {x} + \frac {6 a^{2} c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 a^{2} c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 a^{2} d^{3} x^{\frac {13}{2}}}{13} + \frac {4 a b c^{3} x^{\frac {5}{2}}}{5} + \frac {4 a b c^{2} d x^{\frac {9}{2}}}{3} + \frac {12 a b c d^{2} x^{\frac {13}{2}}}{13} + \frac {4 a b d^{3} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c^{2} d x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c d^{2} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} d^{3} x^{\frac {21}{2}}}{21} \]

[In]

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

[Out]

2*a**2*c**3*sqrt(x) + 6*a**2*c**2*d*x**(5/2)/5 + 2*a**2*c*d**2*x**(9/2)/3 + 2*a**2*d**3*x**(13/2)/13 + 4*a*b*c
**3*x**(5/2)/5 + 4*a*b*c**2*d*x**(9/2)/3 + 12*a*b*c*d**2*x**(13/2)/13 + 4*a*b*d**3*x**(17/2)/17 + 2*b**2*c**3*
x**(9/2)/9 + 6*b**2*c**2*d*x**(13/2)/13 + 6*b**2*c*d**2*x**(17/2)/17 + 2*b**2*d**3*x**(21/2)/21

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{21} \, b^{2} d^{3} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {13}{2}} + 2 \, a^{2} c^{3} \sqrt {x} + \frac {2}{9} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {5}{2}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/21*b^2*d^3*x^(21/2) + 2/17*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(17/2) + 2/13*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x
^(13/2) + 2*a^2*c^3*sqrt(x) + 2/9*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^(9/2) + 2/5*(2*a*b*c^3 + 3*a^2*c^2*d
)*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=\frac {2}{21} \, b^{2} d^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c d^{2} x^{\frac {17}{2}} + \frac {4}{17} \, a b d^{3} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c^{2} d x^{\frac {13}{2}} + \frac {12}{13} \, a b c d^{2} x^{\frac {13}{2}} + \frac {2}{13} \, a^{2} d^{3} x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} c^{3} x^{\frac {9}{2}} + \frac {4}{3} \, a b c^{2} d x^{\frac {9}{2}} + \frac {2}{3} \, a^{2} c d^{2} x^{\frac {9}{2}} + \frac {4}{5} \, a b c^{3} x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c^{2} d x^{\frac {5}{2}} + 2 \, a^{2} c^{3} \sqrt {x} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^3/x^(1/2),x, algorithm="giac")

[Out]

2/21*b^2*d^3*x^(21/2) + 6/17*b^2*c*d^2*x^(17/2) + 4/17*a*b*d^3*x^(17/2) + 6/13*b^2*c^2*d*x^(13/2) + 12/13*a*b*
c*d^2*x^(13/2) + 2/13*a^2*d^3*x^(13/2) + 2/9*b^2*c^3*x^(9/2) + 4/3*a*b*c^2*d*x^(9/2) + 2/3*a^2*c*d^2*x^(9/2) +
4/5*a*b*c^3*x^(5/2) + 6/5*a^2*c^2*d*x^(5/2) + 2*a^2*c^3*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx=x^{9/2}\,\left (\frac {2\,a^2\,c\,d^2}{3}+\frac {4\,a\,b\,c^2\,d}{3}+\frac {2\,b^2\,c^3}{9}\right )+x^{13/2}\,\left (\frac {2\,a^2\,d^3}{13}+\frac {12\,a\,b\,c\,d^2}{13}+\frac {6\,b^2\,c^2\,d}{13}\right )+2\,a^2\,c^3\,\sqrt {x}+\frac {2\,b^2\,d^3\,x^{21/2}}{21}+\frac {2\,a\,c^2\,x^{5/2}\,\left (3\,a\,d+2\,b\,c\right )}{5}+\frac {2\,b\,d^2\,x^{17/2}\,\left (2\,a\,d+3\,b\,c\right )}{17} \]

[In]

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

[Out]

x^(9/2)*((2*b^2*c^3)/9 + (2*a^2*c*d^2)/3 + (4*a*b*c^2*d)/3) + x^(13/2)*((2*a^2*d^3)/13 + (6*b^2*c^2*d)/13 + (1
2*a*b*c*d^2)/13) + 2*a^2*c^3*x^(1/2) + (2*b^2*d^3*x^(21/2))/21 + (2*a*c^2*x^(5/2)*(3*a*d + 2*b*c))/5 + (2*b*d^
2*x^(17/2)*(2*a*d + 3*b*c))/17