\(\int \frac {(a+b x^2)^2 (c+d x^2)}{\sqrt {x}} \, dx\) [395]

Optimal result

Integrand size = 22, antiderivative size = 61 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=2 a^2 c \sqrt {x}+\frac {2}{5} a (2 b c+a d) x^{5/2}+\frac {2}{9} b (b c+2 a d) x^{9/2}+\frac {2}{13} b^2 d x^{13/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=2 a^2 c \sqrt {x}+\frac {2}{9} b x^{9/2} (2 a d+b c)+\frac {2}{5} a x^{5/2} (a d+2 b c)+\frac {2}{13} b^2 d x^{13/2} \]

[In]

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

[Out]

2*a^2*c*Sqrt[x] + (2*a*(2*b*c + a*d)*x^(5/2))/5 + (2*b*(b*c + 2*a*d)*x^(9/2))/9 + (2*b^2*d*x^(13/2))/13

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=\frac {2}{585} \sqrt {x} \left (117 a^2 \left (5 c+d x^2\right )+26 a b x^2 \left (9 c+5 d x^2\right )+5 b^2 x^4 \left (13 c+9 d x^2\right )\right ) \]

[In]

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

[Out]

(2*Sqrt[x]*(117*a^2*(5*c + d*x^2) + 26*a*b*x^2*(9*c + 5*d*x^2) + 5*b^2*x^4*(13*c + 9*d*x^2)))/585

Maple [A] (verified)

Time = 2.68 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {2 b^{2} d \,x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {5}{2}}}{5}+2 a^{2} c \sqrt {x}\)	\(52\)
default	\(\frac {2 b^{2} d \,x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b d +b^{2} c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} d +2 a b c \right ) x^{\frac {5}{2}}}{5}+2 a^{2} c \sqrt {x}\)	\(52\)
trager	\(\left (\frac {2}{13} b^{2} d \,x^{6}+\frac {4}{9} a b d \,x^{4}+\frac {2}{9} b^{2} c \,x^{4}+\frac {2}{5} a^{2} d \,x^{2}+\frac {4}{5} a b c \,x^{2}+2 a^{2} c \right ) \sqrt {x}\)	\(55\)
gosper	\(\frac {2 \sqrt {x}\, \left (45 b^{2} d \,x^{6}+130 a b d \,x^{4}+65 b^{2} c \,x^{4}+117 a^{2} d \,x^{2}+234 a b c \,x^{2}+585 a^{2} c \right )}{585}\)	\(56\)
risch	\(\frac {2 \sqrt {x}\, \left (45 b^{2} d \,x^{6}+130 a b d \,x^{4}+65 b^{2} c \,x^{4}+117 a^{2} d \,x^{2}+234 a b c \,x^{2}+585 a^{2} c \right )}{585}\)	\(56\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=\frac {2}{585} \, {\left (45 \, b^{2} d x^{6} + 65 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + 585 \, a^{2} c + 117 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt {x} \]

[In]

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

[Out]

2/585*(45*b^2*d*x^6 + 65*(b^2*c + 2*a*b*d)*x^4 + 585*a^2*c + 117*(2*a*b*c + a^2*d)*x^2)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=2 a^{2} c \sqrt {x} + \frac {2 a^{2} d x^{\frac {5}{2}}}{5} + \frac {4 a b c x^{\frac {5}{2}}}{5} + \frac {4 a b d x^{\frac {9}{2}}}{9} + \frac {2 b^{2} c x^{\frac {9}{2}}}{9} + \frac {2 b^{2} d x^{\frac {13}{2}}}{13} \]

[In]

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

[Out]

2*a**2*c*sqrt(x) + 2*a**2*d*x**(5/2)/5 + 4*a*b*c*x**(5/2)/5 + 4*a*b*d*x**(9/2)/9 + 2*b**2*c*x**(9/2)/9 + 2*b**
2*d*x**(13/2)/13

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=\frac {2}{13} \, b^{2} d x^{\frac {13}{2}} + \frac {2}{9} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {9}{2}} + 2 \, a^{2} c \sqrt {x} + \frac {2}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {5}{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=\frac {2}{13} \, b^{2} d x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} c x^{\frac {9}{2}} + \frac {4}{9} \, a b d x^{\frac {9}{2}} + \frac {4}{5} \, a b c x^{\frac {5}{2}} + \frac {2}{5} \, a^{2} d x^{\frac {5}{2}} + 2 \, a^{2} c \sqrt {x} \]

[In]

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

[Out]

2/13*b^2*d*x^(13/2) + 2/9*b^2*c*x^(9/2) + 4/9*a*b*d*x^(9/2) + 4/5*a*b*c*x^(5/2) + 2/5*a^2*d*x^(5/2) + 2*a^2*c*
sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{\sqrt {x}} \, dx=x^{5/2}\,\left (\frac {2\,d\,a^2}{5}+\frac {4\,b\,c\,a}{5}\right )+x^{9/2}\,\left (\frac {2\,c\,b^2}{9}+\frac {4\,a\,d\,b}{9}\right )+2\,a^2\,c\,\sqrt {x}+\frac {2\,b^2\,d\,x^{13/2}}{13} \]

[In]

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

[Out]

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