3.4.41 \(\int x^{7/2} (a+b x^2)^3 (A+B x^2) \, dx\)

Optimal. Leaf size=85 \[ \frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{21} b^2 x^{21/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 85, 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}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{9} a^3 A x^{9/2}+\frac {2}{21} b^2 x^{21/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*
B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

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 )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{11/2}+3 a b (A b+a B) x^{15/2}+b^2 (A b+3 a B) x^{19/2}+b^3 B x^{23/2}\right ) \, dx\\ &=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 (3 A b+a B) x^{13/2}+\frac {6}{17} a b (A b+a B) x^{17/2}+\frac {2}{21} b^2 (A b+3 a B) x^{21/2}+\frac {2}{25} b^3 B x^{25/2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{21} b^2 x^{21/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*
B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

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IntegrateAlgebraic [A]  time = 0.05, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (38675 a^3 A x^{9/2}+26775 a^3 B x^{13/2}+80325 a^2 A b x^{13/2}+61425 a^2 b B x^{17/2}+61425 a A b^2 x^{17/2}+49725 a b^2 B x^{21/2}+16575 A b^3 x^{21/2}+13923 b^3 B x^{25/2}\right )}{348075} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*(38675*a^3*A*x^(9/2) + 80325*a^2*A*b*x^(13/2) + 26775*a^3*B*x^(13/2) + 61425*a*A*b^2*x^(17/2) + 61425*a^2*b
*B*x^(17/2) + 16575*A*b^3*x^(21/2) + 49725*a*b^2*B*x^(21/2) + 13923*b^3*B*x^(25/2)))/348075

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fricas [A]  time = 1.23, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{348075} \, {\left (13923 \, B b^{3} x^{12} + 16575 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{10} + 61425 \, {\left (B a^{2} b + A a b^{2}\right )} x^{8} + 38675 \, A a^{3} x^{4} + 26775 \, {\left (B a^{3} + 3 \, A a^{2} b\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)^3*(B*x^2+A),x, algorithm="fricas")

[Out]

2/348075*(13923*B*b^3*x^12 + 16575*(3*B*a*b^2 + A*b^3)*x^10 + 61425*(B*a^2*b + A*a*b^2)*x^8 + 38675*A*a^3*x^4
+ 26775*(B*a^3 + 3*A*a^2*b)*x^6)*sqrt(x)

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giac [A]  time = 0.37, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{7} \, B a b^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A b^{3} x^{\frac {21}{2}} + \frac {6}{17} \, B a^{2} b x^{\frac {17}{2}} + \frac {6}{17} \, A a b^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B a^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A a^{2} b x^{\frac {13}{2}} + \frac {2}{9} \, A a^{3} 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)^3*(B*x^2+A),x, algorithm="giac")

[Out]

2/25*B*b^3*x^(25/2) + 2/7*B*a*b^2*x^(21/2) + 2/21*A*b^3*x^(21/2) + 6/17*B*a^2*b*x^(17/2) + 6/17*A*a*b^2*x^(17/
2) + 2/13*B*a^3*x^(13/2) + 6/13*A*a^2*b*x^(13/2) + 2/9*A*a^3*x^(9/2)

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maple [A]  time = 0.01, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (13923 B \,b^{3} x^{8}+16575 x^{6} A \,b^{3}+49725 B a \,b^{2} x^{6}+61425 x^{4} A a \,b^{2}+61425 x^{4} B \,a^{2} b +80325 A \,a^{2} b \,x^{2}+26775 B \,a^{3} x^{2}+38675 A \,a^{3}\right ) x^{\frac {9}{2}}}{348075} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x)

[Out]

2/348075*x^(9/2)*(13923*B*b^3*x^8+16575*A*b^3*x^6+49725*B*a*b^2*x^6+61425*A*a*b^2*x^4+61425*B*a^2*b*x^4+80325*
A*a^2*b*x^2+26775*B*a^3*x^2+38675*A*a^3)

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maxima [A]  time = 1.14, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {21}{2}} + \frac {6}{17} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B a^{3} + 3 \, A a^{2} b\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)^3*(B*x^2+A),x, algorithm="maxima")

[Out]

2/25*B*b^3*x^(25/2) + 2/21*(3*B*a*b^2 + A*b^3)*x^(21/2) + 6/17*(B*a^2*b + A*a*b^2)*x^(17/2) + 2/9*A*a^3*x^(9/2
) + 2/13*(B*a^3 + 3*A*a^2*b)*x^(13/2)

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mupad [B]  time = 0.04, size = 69, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,a^3}{13}+\frac {6\,A\,b\,a^2}{13}\right )+x^{21/2}\,\left (\frac {2\,A\,b^3}{21}+\frac {2\,B\,a\,b^2}{7}\right )+\frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,b^3\,x^{25/2}}{25}+\frac {6\,a\,b\,x^{17/2}\,\left (A\,b+B\,a\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(A + B*x^2)*(a + b*x^2)^3,x)

[Out]

x^(13/2)*((2*B*a^3)/13 + (6*A*a^2*b)/13) + x^(21/2)*((2*A*b^3)/21 + (2*B*a*b^2)/7) + (2*A*a^3*x^(9/2))/9 + (2*
B*b^3*x^(25/2))/25 + (6*a*b*x^(17/2)*(A*b + B*a))/17

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sympy [A]  time = 34.25, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {6 A a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 A a b^{2} x^{\frac {17}{2}}}{17} + \frac {2 A b^{3} x^{\frac {21}{2}}}{21} + \frac {2 B a^{3} x^{\frac {13}{2}}}{13} + \frac {6 B a^{2} b x^{\frac {17}{2}}}{17} + \frac {2 B a b^{2} x^{\frac {21}{2}}}{7} + \frac {2 B b^{3} x^{\frac {25}{2}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(13/2)/13 + 6*A*a*b**2*x**(17/2)/17 + 2*A*b**3*x**(21/2)/21 + 2*B*a**3*x**
(13/2)/13 + 6*B*a**2*b*x**(17/2)/17 + 2*B*a*b**2*x**(21/2)/7 + 2*B*b**3*x**(25/2)/25

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