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

Optimal. Leaf size=37 \[ \frac {2}{5} x^{5/2} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{11} b B x^{11/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, 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}{5} x^{5/2} (a B+A b)-\frac {2 a A}{\sqrt {x}}+\frac {2}{11} b B x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*A)/Sqrt[x] + (2*(A*b + a*B)*x^(5/2))/5 + (2*b*B*x^(11/2))/11

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

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.95 \begin {gather*} \frac {2 \left (-55 a A+11 a B x^3+11 A b x^3+5 b B x^6\right )}{55 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-55*a*A + 11*A*b*x^3 + 11*a*B*x^3 + 5*b*B*x^6))/(55*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.02, size = 35, normalized size = 0.95 \begin {gather*} \frac {2 \left (-55 a A+11 a B x^3+11 A b x^3+5 b B x^6\right )}{55 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-55*a*A + 11*A*b*x^3 + 11*a*B*x^3 + 5*b*B*x^6))/(55*Sqrt[x])

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fricas [A]  time = 0.78, size = 29, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (5 \, B b x^{6} + 11 \, {\left (B a + A b\right )} x^{3} - 55 \, A a\right )}}{55 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/55*(5*B*b*x^6 + 11*(B*a + A*b)*x^3 - 55*A*a)/sqrt(x)

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giac [A]  time = 0.15, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{11} \, B b x^{\frac {11}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} - \frac {2 \, A a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*b*x^(11/2) + 2/5*B*a*x^(5/2) + 2/5*A*b*x^(5/2) - 2*A*a/sqrt(x)

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maple [A]  time = 0.04, size = 32, normalized size = 0.86 \begin {gather*} -\frac {2 \left (-5 B b \,x^{6}-11 A b \,x^{3}-11 B a \,x^{3}+55 A a \right )}{55 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/55*(-5*B*b*x^6-11*A*b*x^3-11*B*a*x^3+55*A*a)/x^(1/2)

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maxima [A]  time = 0.46, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{11} \, B b x^{\frac {11}{2}} + \frac {2}{5} \, {\left (B a + A b\right )} x^{\frac {5}{2}} - \frac {2 \, A a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*b*x^(11/2) + 2/5*(B*a + A*b)*x^(5/2) - 2*A*a/sqrt(x)

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mupad [B]  time = 2.60, size = 31, normalized size = 0.84 \begin {gather*} \frac {22\,A\,b\,x^3-110\,A\,a+22\,B\,a\,x^3+10\,B\,b\,x^6}{55\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(22*A*b*x^3 - 110*A*a + 22*B*a*x^3 + 10*B*b*x^6)/(55*x^(1/2))

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sympy [A]  time = 2.77, size = 44, normalized size = 1.19 \begin {gather*} - \frac {2 A a}{\sqrt {x}} + \frac {2 A b x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {11}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a/sqrt(x) + 2*A*b*x**(5/2)/5 + 2*B*a*x**(5/2)/5 + 2*B*b*x**(11/2)/11

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