3.5.3 \(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{\sqrt {x}} \, dx\) [403]

Optimal. Leaf size=95 \[ 2 a^2 c^2 \sqrt {x}+\frac {4}{5} a c (b c+a d) x^{5/2}+\frac {2}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{9/2}+\frac {4}{13} b d (b c+a d) x^{13/2}+\frac {2}{17} b^2 d^2 x^{17/2} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459} \begin {gather*} \frac {2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt {x}+\frac {4}{13} b d x^{13/2} (a d+b c)+\frac {4}{5} a c x^{5/2} (a d+b c)+\frac {2}{17} b^2 d^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*c^2*Sqrt[x] + (4*a*c*(b*c + a*d)*x^(5/2))/5 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(9/2))/9 + (4*b*d*(b*
c + a*d)*x^(13/2))/13 + (2*b^2*d^2*x^(17/2))/17

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

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Mathematica [A]
time = 0.06, size = 93, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {x} \left (221 a^2 \left (45 c^2+18 c d x^2+5 d^2 x^4\right )+34 a b x^2 \left (117 c^2+130 c d x^2+45 d^2 x^4\right )+5 b^2 x^4 \left (221 c^2+306 c d x^2+117 d^2 x^4\right )\right )}{9945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(221*a^2*(45*c^2 + 18*c*d*x^2 + 5*d^2*x^4) + 34*a*b*x^2*(117*c^2 + 130*c*d*x^2 + 45*d^2*x^4) + 5*b^
2*x^4*(221*c^2 + 306*c*d*x^2 + 117*d^2*x^4)))/9945

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Maple [A]
time = 0.10, size = 90, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{\frac {5}{2}}}{5}+2 a^{2} c^{2} \sqrt {x}\)	\(90\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{\frac {5}{2}}}{5}+2 a^{2} c^{2} \sqrt {x}\)	\(90\)
trager	\(\left (\frac {2}{17} b^{2} d^{2} x^{8}+\frac {4}{13} a b \,d^{2} x^{6}+\frac {4}{13} b^{2} c d \,x^{6}+\frac {2}{9} a^{2} d^{2} x^{4}+\frac {8}{9} a b c d \,x^{4}+\frac {2}{9} b^{2} c^{2} x^{4}+\frac {4}{5} a^{2} c d \,x^{2}+\frac {4}{5} a b \,c^{2} x^{2}+2 a^{2} c^{2}\right ) \sqrt {x}\)	\(96\)
gosper	\(\frac {2 \sqrt {x}\, \left (585 b^{2} d^{2} x^{8}+1530 a b \,d^{2} x^{6}+1530 b^{2} c d \,x^{6}+1105 a^{2} d^{2} x^{4}+4420 a b c d \,x^{4}+1105 b^{2} c^{2} x^{4}+3978 a^{2} c d \,x^{2}+3978 a b \,c^{2} x^{2}+9945 a^{2} c^{2}\right )}{9945}\)	\(97\)
risch	\(\frac {2 \sqrt {x}\, \left (585 b^{2} d^{2} x^{8}+1530 a b \,d^{2} x^{6}+1530 b^{2} c d \,x^{6}+1105 a^{2} d^{2} x^{4}+4420 a b c d \,x^{4}+1105 b^{2} c^{2} x^{4}+3978 a^{2} c d \,x^{2}+3978 a b \,c^{2} x^{2}+9945 a^{2} c^{2}\right )}{9945}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 85, normalized size = 0.89 \begin {gather*} \frac {2}{17} \, b^{2} d^{2} x^{\frac {17}{2}} + \frac {4}{13} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {9}{2}} + 2 \, a^{2} c^{2} \sqrt {x} + \frac {4}{5} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.70, size = 87, normalized size = 0.92 \begin {gather*} \frac {2}{9945} \, {\left (585 \, b^{2} d^{2} x^{8} + 1530 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 1105 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + 9945 \, a^{2} c^{2} + 3978 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9945*(585*b^2*d^2*x^8 + 1530*(b^2*c*d + a*b*d^2)*x^6 + 1105*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 9945*a^2*c
^2 + 3978*(a*b*c^2 + a^2*c*d)*x^2)*sqrt(x)

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Sympy [A]
time = 0.55, size = 134, normalized size = 1.41 \begin {gather*} 2 a^{2} c^{2} \sqrt {x} + \frac {4 a^{2} c d x^{\frac {5}{2}}}{5} + \frac {2 a^{2} d^{2} x^{\frac {9}{2}}}{9} + \frac {4 a b c^{2} x^{\frac {5}{2}}}{5} + \frac {8 a b c d x^{\frac {9}{2}}}{9} + \frac {4 a b d^{2} x^{\frac {13}{2}}}{13} + \frac {2 b^{2} c^{2} x^{\frac {9}{2}}}{9} + \frac {4 b^{2} c d x^{\frac {13}{2}}}{13} + \frac {2 b^{2} d^{2} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*c**2*sqrt(x) + 4*a**2*c*d*x**(5/2)/5 + 2*a**2*d**2*x**(9/2)/9 + 4*a*b*c**2*x**(5/2)/5 + 8*a*b*c*d*x**(9
/2)/9 + 4*a*b*d**2*x**(13/2)/13 + 2*b**2*c**2*x**(9/2)/9 + 4*b**2*c*d*x**(13/2)/13 + 2*b**2*d**2*x**(17/2)/17

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Giac [A]
time = 1.12, size = 94, normalized size = 0.99 \begin {gather*} \frac {2}{17} \, b^{2} d^{2} x^{\frac {17}{2}} + \frac {4}{13} \, b^{2} c d x^{\frac {13}{2}} + \frac {4}{13} \, a b d^{2} x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} c^{2} x^{\frac {9}{2}} + \frac {8}{9} \, a b c d x^{\frac {9}{2}} + \frac {2}{9} \, a^{2} d^{2} x^{\frac {9}{2}} + \frac {4}{5} \, a b c^{2} x^{\frac {5}{2}} + \frac {4}{5} \, a^{2} c d x^{\frac {5}{2}} + 2 \, a^{2} c^{2} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*b^2*d^2*x^(17/2) + 4/13*b^2*c*d*x^(13/2) + 4/13*a*b*d^2*x^(13/2) + 2/9*b^2*c^2*x^(9/2) + 8/9*a*b*c*d*x^(9
/2) + 2/9*a^2*d^2*x^(9/2) + 4/5*a*b*c^2*x^(5/2) + 4/5*a^2*c*d*x^(5/2) + 2*a^2*c^2*sqrt(x)

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Mupad [B]
time = 0.02, size = 78, normalized size = 0.82 \begin {gather*} x^{9/2}\,\left (\frac {2\,a^2\,d^2}{9}+\frac {8\,a\,b\,c\,d}{9}+\frac {2\,b^2\,c^2}{9}\right )+2\,a^2\,c^2\,\sqrt {x}+\frac {2\,b^2\,d^2\,x^{17/2}}{17}+\frac {4\,a\,c\,x^{5/2}\,\left (a\,d+b\,c\right )}{5}+\frac {4\,b\,d\,x^{13/2}\,\left (a\,d+b\,c\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(9/2)*((2*a^2*d^2)/9 + (2*b^2*c^2)/9 + (8*a*b*c*d)/9) + 2*a^2*c^2*x^(1/2) + (2*b^2*d^2*x^(17/2))/17 + (4*a*c
*x^(5/2)*(a*d + b*c))/5 + (4*b*d*x^(13/2)*(a*d + b*c))/13

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