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

Optimal. Leaf size=39 \[ \frac {2}{7} x^{7/2} (a B+A b)+\frac {2}{5} a A x^{5/2}+\frac {2}{9} b B x^{9/2} \]

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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} \frac {2}{7} x^{7/2} (a B+A b)+\frac {2}{5} a A x^{5/2}+\frac {2}{9} b B x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x)*(A + B*x),x]

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(9/2))/9

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {2}{315} x^{5/2} (9 a (7 A+5 B x)+5 b x (9 A+7 B x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(9*a*(7*A + 5*B*x) + 5*b*x*(9*A + 7*B*x)))/315

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 1.05 \begin {gather*} \frac {2}{315} \left (63 a A x^{5/2}+45 a B x^{7/2}+45 A b x^{7/2}+35 b B x^{9/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(63*a*A*x^(5/2) + 45*A*b*x^(7/2) + 45*a*B*x^(7/2) + 35*b*B*x^(9/2)))/315

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fricas [A]  time = 0.97, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{315} \, {\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \, {\left (B a + A b\right )} x^{3}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*b*x^4 + 63*A*a*x^2 + 45*(B*a + A*b)*x^3)*sqrt(x)

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giac [A]  time = 1.22, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{7} \, B a x^{\frac {7}{2}} + \frac {2}{7} \, A b x^{\frac {7}{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+a)*(B*x+A),x, algorithm="giac")

[Out]

2/9*B*b*x^(9/2) + 2/7*B*a*x^(7/2) + 2/7*A*b*x^(7/2) + 2/5*A*a*x^(5/2)

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maple [A]  time = 0.00, size = 28, normalized size = 0.72 \begin {gather*} \frac {2 \left (35 B b \,x^{2}+45 A b x +45 B a x +63 A a \right ) x^{\frac {5}{2}}}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*x^(5/2)*(35*B*b*x^2+45*A*b*x+45*B*a*x+63*A*a)

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maxima [A]  time = 0.88, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{5} \, A a x^{\frac {5}{2}} + \frac {2}{7} \, {\left (B a + A b\right )} x^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*b*x^(9/2) + 2/5*A*a*x^(5/2) + 2/7*(B*a + A*b)*x^(7/2)

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mupad [B]  time = 0.05, size = 27, normalized size = 0.69 \begin {gather*} \frac {2\,x^{5/2}\,\left (63\,A\,a+45\,A\,b\,x+45\,B\,a\,x+35\,B\,b\,x^2\right )}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(A + B*x)*(a + b*x),x)

[Out]

(2*x^(5/2)*(63*A*a + 45*A*b*x + 45*B*a*x + 35*B*b*x^2))/315

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sympy [A]  time = 1.01, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A a x^{\frac {5}{2}}}{5} + \frac {2 A b x^{\frac {7}{2}}}{7} + \frac {2 B a x^{\frac {7}{2}}}{7} + \frac {2 B b x^{\frac {9}{2}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a*x**(5/2)/5 + 2*A*b*x**(7/2)/7 + 2*B*a*x**(7/2)/7 + 2*B*b*x**(9/2)/9

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