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

Optimal. Leaf size=61 \[ 2 a^2 c \sqrt {x}+\frac {2}{9} b x^{9/2} (2 a d+b c)+\frac {2}{5} a x^{5/2} (a d+2 b c)+\frac {2}{13} b^2 d x^{13/2} \]

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \begin {gather*} 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} \end {gather*}

Antiderivative was successfully verified.

[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 448

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 )}{\sqrt {x}} \, dx &=\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*}

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Mathematica [A]  time = 0.03, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{585} \sqrt {x} \left (585 a^2 c+65 b x^4 (2 a d+b c)+117 a x^2 (a d+2 b c)+45 b^2 d x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.03, size = 69, normalized size = 1.13 \begin {gather*} \frac {2}{585} \left (585 a^2 c \sqrt {x}+117 a^2 d x^{5/2}+234 a b c x^{5/2}+130 a b d x^{9/2}+65 b^2 c x^{9/2}+45 b^2 d x^{13/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(585*a^2*c*Sqrt[x] + 234*a*b*c*x^(5/2) + 117*a^2*d*x^(5/2) + 65*b^2*c*x^(9/2) + 130*a*b*d*x^(9/2) + 45*b^2*
d*x^(13/2)))/585

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fricas [A]  time = 1.27, size = 53, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.28, size = 53, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.01, size = 56, normalized size = 0.92 \begin {gather*} \frac {2 \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 ) \sqrt {x}}{585} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*x^(1/2)*(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)

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maxima [A]  time = 1.00, size = 51, normalized size = 0.84 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.05, size = 51, normalized size = 0.84 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 2.12, size = 78, normalized size = 1.28 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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