3.4.73 \(\int x^{7/2} (a+b x^2)^2 (c+d x^2) \, dx\)

Optimal. Leaf size=63 \[ \frac {2}{9} a^2 c x^{9/2}+\frac {2}{17} b x^{17/2} (2 a d+b c)+\frac {2}{13} a x^{13/2} (a d+2 b c)+\frac {2}{21} b^2 d x^{21/2} \]

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Rubi [A]  time = 0.03, antiderivative size = 63, 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*} \frac {2}{9} a^2 c x^{9/2}+\frac {2}{17} b x^{17/2} (2 a d+b c)+\frac {2}{13} a x^{13/2} (a d+2 b c)+\frac {2}{21} b^2 d x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 53, normalized size = 0.84 \begin {gather*} \frac {2 x^{9/2} \left (1547 a^2 c+819 b x^4 (2 a d+b c)+1071 a x^2 (a d+2 b c)+663 b^2 d x^6\right )}{13923} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(2*x^(9/2)*(1547*a^2*c + 1071*a*(2*b*c + a*d)*x^2 + 819*b*(b*c + 2*a*d)*x^4 + 663*b^2*d*x^6))/13923

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IntegrateAlgebraic [A]  time = 0.04, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (1547 a^2 c x^{9/2}+1071 a^2 d x^{13/2}+2142 a b c x^{13/2}+1638 a b d x^{17/2}+819 b^2 c x^{17/2}+663 b^2 d x^{21/2}\right )}{13923} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(2*(1547*a^2*c*x^(9/2) + 2142*a*b*c*x^(13/2) + 1071*a^2*d*x^(13/2) + 819*b^2*c*x^(17/2) + 1638*a*b*d*x^(17/2)
+ 663*b^2*d*x^(21/2)))/13923

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fricas [A]  time = 1.05, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{13923} \, {\left (663 \, b^{2} d x^{10} + 819 \, {\left (b^{2} c + 2 \, a b d\right )} x^{8} + 1547 \, a^{2} c x^{4} + 1071 \, {\left (2 \, a b c + a^{2} d\right )} x^{6}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x, algorithm="fricas")

[Out]

2/13923*(663*b^2*d*x^10 + 819*(b^2*c + 2*a*b*d)*x^8 + 1547*a^2*c*x^4 + 1071*(2*a*b*c + a^2*d)*x^6)*sqrt(x)

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giac [A]  time = 0.40, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{21} \, b^{2} d x^{\frac {21}{2}} + \frac {2}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {4}{17} \, a b d x^{\frac {17}{2}} + \frac {4}{13} \, a b c x^{\frac {13}{2}} + \frac {2}{13} \, a^{2} d x^{\frac {13}{2}} + \frac {2}{9} \, a^{2} c x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 56, normalized size = 0.89 \begin {gather*} \frac {2 \left (663 b^{2} d \,x^{6}+1638 a b d \,x^{4}+819 b^{2} c \,x^{4}+1071 a^{2} d \,x^{2}+2142 a b c \,x^{2}+1547 a^{2} c \right ) x^{\frac {9}{2}}}{13923} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x)

[Out]

2/13923*x^(9/2)*(663*b^2*d*x^6+1638*a*b*d*x^4+819*b^2*c*x^4+1071*a^2*d*x^2+2142*a*b*c*x^2+1547*a^2*c)

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maxima [A]  time = 1.00, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{21} \, b^{2} d x^{\frac {21}{2}} + \frac {2}{17} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {17}{2}} + \frac {2}{9} \, a^{2} c x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {13}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 51, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,d\,a^2}{13}+\frac {4\,b\,c\,a}{13}\right )+x^{17/2}\,\left (\frac {2\,c\,b^2}{17}+\frac {4\,a\,d\,b}{17}\right )+\frac {2\,a^2\,c\,x^{9/2}}{9}+\frac {2\,b^2\,d\,x^{21/2}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x)

[Out]

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

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sympy [A]  time = 19.90, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 a^{2} c x^{\frac {9}{2}}}{9} + \frac {2 a^{2} d x^{\frac {13}{2}}}{13} + \frac {4 a b c x^{\frac {13}{2}}}{13} + \frac {4 a b d x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 b^{2} d x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

2*a**2*c*x**(9/2)/9 + 2*a**2*d*x**(13/2)/13 + 4*a*b*c*x**(13/2)/13 + 4*a*b*d*x**(17/2)/17 + 2*b**2*c*x**(17/2)
/17 + 2*b**2*d*x**(21/2)/21

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