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

Optimal. Leaf size=39 \[ \frac {2}{15} x^{15/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{21} b B x^{21/2} \]

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Rubi [A]  time = 0.02, 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 = {448} \begin {gather*} \frac {2}{15} x^{15/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{21} b B x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*A*x^(9/2))/9 + (2*(A*b + a*B)*x^(15/2))/15 + (2*b*B*x^(21/2))/21

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

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{315} x^{9/2} \left (21 x^3 (a B+A b)+35 a A+15 b B x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(35*a*A + 21*(A*b + a*B)*x^3 + 15*b*B*x^6))/315

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IntegrateAlgebraic [A]  time = 0.02, size = 35, normalized size = 0.90 \begin {gather*} \frac {2}{315} x^{9/2} \left (35 a A+21 a B x^3+21 A b x^3+15 b B x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(35*a*A + 21*A*b*x^3 + 21*a*B*x^3 + 15*b*B*x^6))/315

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fricas [A]  time = 0.88, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{315} \, {\left (15 \, B b x^{10} + 21 \, {\left (B a + A b\right )} x^{7} + 35 \, A a x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)*(B*x^3+A),x, algorithm="fricas")

[Out]

2/315*(15*B*b*x^10 + 21*(B*a + A*b)*x^7 + 35*A*a*x^4)*sqrt(x)

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giac [A]  time = 0.17, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{21} \, B b x^{\frac {21}{2}} + \frac {2}{15} \, B a x^{\frac {15}{2}} + \frac {2}{15} \, A b x^{\frac {15}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)*(B*x^3+A),x, algorithm="giac")

[Out]

2/21*B*b*x^(21/2) + 2/15*B*a*x^(15/2) + 2/15*A*b*x^(15/2) + 2/9*A*a*x^(9/2)

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maple [A]  time = 0.05, size = 32, normalized size = 0.82 \begin {gather*} \frac {2 \left (15 B b \,x^{6}+21 A b \,x^{3}+21 B a \,x^{3}+35 A a \right ) x^{\frac {9}{2}}}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*x^(9/2)*(15*B*b*x^6+21*A*b*x^3+21*B*a*x^3+35*A*a)

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maxima [A]  time = 0.45, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, B b x^{\frac {21}{2}} + \frac {2}{15} \, {\left (B a + A b\right )} x^{\frac {15}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^3+a)*(B*x^3+A),x, algorithm="maxima")

[Out]

2/21*B*b*x^(21/2) + 2/15*(B*a + A*b)*x^(15/2) + 2/9*A*a*x^(9/2)

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mupad [B]  time = 0.05, size = 31, normalized size = 0.79 \begin {gather*} \frac {2\,x^{9/2}\,\left (35\,A\,a+21\,A\,b\,x^3+21\,B\,a\,x^3+15\,B\,b\,x^6\right )}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^(9/2)*(35*A*a + 21*A*b*x^3 + 21*B*a*x^3 + 15*B*b*x^6))/315

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sympy [A]  time = 20.86, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {15}{2}}}{15} + \frac {2 B a x^{\frac {15}{2}}}{15} + \frac {2 B b x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(9/2)/9 + 2*A*b*x**(15/2)/15 + 2*B*a*x**(15/2)/15 + 2*B*b*x**(21/2)/21

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