3.4.45 \(\int x^{3/2} (a+b x^2) (A+B x^2) \, dx\) [345]

Optimal. Leaf size=39 \[ \frac {2}{5} a A x^{5/2}+\frac {2}{9} (A b+a B) x^{9/2}+\frac {2}{13} b B x^{13/2} \]

[Out]

2/5*a*A*x^(5/2)+2/9*(A*b+B*a)*x^(9/2)+2/13*b*B*x^(13/2)

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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \begin {gather*} \frac {2}{9} x^{9/2} (a B+A b)+\frac {2}{5} a A x^{5/2}+\frac {2}{13} b B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(9/2))/9 + (2*b*B*x^(13/2))/13

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^{3/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A x^{3/2}+(A b+a B) x^{7/2}+b B x^{11/2}\right ) \, dx\\ &=\frac {2}{5} a A x^{5/2}+\frac {2}{9} (A b+a B) x^{9/2}+\frac {2}{13} b B x^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.90 \begin {gather*} \frac {2}{585} x^{5/2} \left (117 a A+65 A b x^2+65 a B x^2+45 b B x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(117*a*A + 65*A*b*x^2 + 65*a*B*x^2 + 45*b*B*x^4))/585

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Maple [A]
time = 0.09, size = 28, normalized size = 0.72

method	result	size
derivativedivides	\(\frac {2 a A \,x^{\frac {5}{2}}}{5}+\frac {2 \left (A b +B a \right ) x^{\frac {9}{2}}}{9}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\)	\(28\)
default	\(\frac {2 a A \,x^{\frac {5}{2}}}{5}+\frac {2 \left (A b +B a \right ) x^{\frac {9}{2}}}{9}+\frac {2 b B \,x^{\frac {13}{2}}}{13}\)	\(28\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (45 b B \,x^{4}+65 A b \,x^{2}+65 B a \,x^{2}+117 A a \right )}{585}\)	\(32\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (45 b B \,x^{4}+65 A b \,x^{2}+65 B a \,x^{2}+117 A a \right )}{585}\)	\(32\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (45 b B \,x^{4}+65 A b \,x^{2}+65 B a \,x^{2}+117 A a \right )}{585}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

2/5*a*A*x^(5/2)+2/9*(A*b+B*a)*x^(9/2)+2/13*b*B*x^(13/2)

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Maxima [A]
time = 0.32, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{9} \, {\left (B a + A b\right )} x^{\frac {9}{2}} + \frac {2}{5} \, A a x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="maxima")

[Out]

2/13*B*b*x^(13/2) + 2/9*(B*a + A*b)*x^(9/2) + 2/5*A*a*x^(5/2)

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Fricas [A]
time = 0.97, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{585} \, {\left (45 \, B b x^{6} + 65 \, {\left (B a + A b\right )} x^{4} + 117 \, A a x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="fricas")

[Out]

2/585*(45*B*b*x^6 + 65*(B*a + A*b)*x^4 + 117*A*a*x^2)*sqrt(x)

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Sympy [A]
time = 0.30, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A a x^{\frac {5}{2}}}{5} + \frac {2 A b x^{\frac {9}{2}}}{9} + \frac {2 B a x^{\frac {9}{2}}}{9} + \frac {2 B b x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(5/2)/5 + 2*A*b*x**(9/2)/9 + 2*B*a*x**(9/2)/9 + 2*B*b*x**(13/2)/13

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Giac [A]
time = 0.84, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{9} \, B a x^{\frac {9}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} + \frac {2}{5} \, A a x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="giac")

[Out]

2/13*B*b*x^(13/2) + 2/9*B*a*x^(9/2) + 2/9*A*b*x^(9/2) + 2/5*A*a*x^(5/2)

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Mupad [B]
time = 0.02, size = 31, normalized size = 0.79 \begin {gather*} \frac {2\,x^{5/2}\,\left (117\,A\,a+65\,A\,b\,x^2+65\,B\,a\,x^2+45\,B\,b\,x^4\right )}{585} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^(5/2)*(117*A*a + 65*A*b*x^2 + 65*B*a*x^2 + 45*B*b*x^4))/585

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