3.4.51 \(\int x^{7/2} (a+b x^2)^2 (A+B x^2) \, dx\) [351]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {459} \begin {gather*} \frac {2}{9} a^2 A x^{9/2}+\frac {2}{17} b x^{17/2} (2 a B+A b)+\frac {2}{13} a x^{13/2} (a B+2 A b)+\frac {2}{21} b^2 B x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(13/2))/13 + (2*b*(A*b + 2*a*B)*x^(17/2))/17 + (2*b^2*B*x^(21/2))/2
1

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

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Mathematica [A]
time = 0.04, size = 60, normalized size = 0.95 \begin {gather*} \frac {2 x^{9/2} \left (119 a^2 \left (13 A+9 B x^2\right )+126 a b x^2 \left (17 A+13 B x^2\right )+39 b^2 x^4 \left (21 A+17 B x^2\right )\right )}{13923} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(119*a^2*(13*A + 9*B*x^2) + 126*a*b*x^2*(17*A + 13*B*x^2) + 39*b^2*x^4*(21*A + 17*B*x^2)))/13923

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Maple [A]
time = 0.10, size = 52, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {2 b^{2} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\)	\(52\)
default	\(\frac {2 b^{2} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{2} A \,x^{\frac {9}{2}}}{9}\)	\(52\)
gosper	\(\frac {2 x^{\frac {9}{2}} \left (663 b^{2} B \,x^{6}+819 A \,b^{2} x^{4}+1638 B a b \,x^{4}+2142 a A b \,x^{2}+1071 B \,a^{2} x^{2}+1547 a^{2} A \right )}{13923}\)	\(56\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (663 b^{2} B \,x^{6}+819 A \,b^{2} x^{4}+1638 B a b \,x^{4}+2142 a A b \,x^{2}+1071 B \,a^{2} x^{2}+1547 a^{2} A \right )}{13923}\)	\(56\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (663 b^{2} B \,x^{6}+819 A \,b^{2} x^{4}+1638 B a b \,x^{4}+2142 a A b \,x^{2}+1071 B \,a^{2} x^{2}+1547 a^{2} A \right )}{13923}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{21} \, B b^{2} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B a^{2} + 2 \, A a 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)^2*(B*x^2+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.04, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{13923} \, {\left (663 \, B b^{2} x^{10} + 819 \, {\left (2 \, B a b + A b^{2}\right )} x^{8} + 1547 \, A a^{2} x^{4} + 1071 \, {\left (B a^{2} + 2 \, A a 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)^2*(B*x^2+A),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.27, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a b x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} x^{\frac {17}{2}}}{17} + \frac {2 B a^{2} x^{\frac {13}{2}}}{13} + \frac {4 B a b x^{\frac {17}{2}}}{17} + \frac {2 B b^{2} 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*(B*x**2+A),x)

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(13/2)/13 + 2*A*b**2*x**(17/2)/17 + 2*B*a**2*x**(13/2)/13 + 4*B*a*b*x**(17/2)
/17 + 2*B*b**2*x**(21/2)/21

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Giac [A]
time = 0.98, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{21} \, B b^{2} x^{\frac {21}{2}} + \frac {4}{17} \, B a b x^{\frac {17}{2}} + \frac {2}{17} \, A b^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B a^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A a b x^{\frac {13}{2}} + \frac {2}{9} \, A a^{2} 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*(B*x^2+A),x, algorithm="giac")

[Out]

2/21*B*b^2*x^(21/2) + 4/17*B*a*b*x^(17/2) + 2/17*A*b^2*x^(17/2) + 2/13*B*a^2*x^(13/2) + 4/13*A*a*b*x^(13/2) +
2/9*A*a^2*x^(9/2)

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Mupad [B]
time = 0.11, size = 51, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,a^2}{13}+\frac {4\,A\,b\,a}{13}\right )+x^{17/2}\,\left (\frac {2\,A\,b^2}{17}+\frac {4\,B\,a\,b}{17}\right )+\frac {2\,A\,a^2\,x^{9/2}}{9}+\frac {2\,B\,b^2\,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)*(a + b*x^2)^2,x)

[Out]

x^(13/2)*((2*B*a^2)/13 + (4*A*a*b)/13) + x^(17/2)*((2*A*b^2)/17 + (4*B*a*b)/17) + (2*A*a^2*x^(9/2))/9 + (2*B*b
^2*x^(21/2))/21

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