3.2.31 \(\int \frac {(a+b x^3) (A+B x^3)}{\sqrt {x}} \, dx\)

Optimal. Leaf size=37 \[ \frac {2}{7} x^{7/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{13} b B x^{13/2} \]

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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}{7} x^{7/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{13} b B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*A*Sqrt[x] + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(13/2))/13

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

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Mathematica [A]  time = 0.07, size = 33, normalized size = 0.89 \begin {gather*} \frac {2}{91} \sqrt {x} \left (13 x^3 (a B+A b)+91 a A+7 b B x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(91*a*A + 13*(A*b + a*B)*x^3 + 7*b*B*x^6))/91

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IntegrateAlgebraic [A]  time = 0.02, size = 41, normalized size = 1.11 \begin {gather*} \frac {2}{91} \left (91 a A \sqrt {x}+13 a B x^{7/2}+13 A b x^{7/2}+7 b B x^{13/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(91*a*A*Sqrt[x] + 13*A*b*x^(7/2) + 13*a*B*x^(7/2) + 7*b*B*x^(13/2)))/91

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fricas [A]  time = 0.75, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{91} \, {\left (7 \, B b x^{6} + 13 \, {\left (B a + A b\right )} x^{3} + 91 \, A a\right )} \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^(1/2),x, algorithm="fricas")

[Out]

2/91*(7*B*b*x^6 + 13*(B*a + A*b)*x^3 + 91*A*a)*sqrt(x)

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giac [A]  time = 0.15, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{7} \, B a x^{\frac {7}{2}} + \frac {2}{7} \, A b x^{\frac {7}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/13*B*b*x^(13/2) + 2/7*B*a*x^(7/2) + 2/7*A*b*x^(7/2) + 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 (7 B b \,x^{6}+13 A b \,x^{3}+13 B a \,x^{3}+91 A a \right ) \sqrt {x}}{91} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(B*x^3+A)/x^(1/2),x)

[Out]

2/91*x^(1/2)*(7*B*b*x^6+13*A*b*x^3+13*B*a*x^3+91*A*a)

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maxima [A]  time = 0.50, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{7} \, {\left (B a + A b\right )} x^{\frac {7}{2}} + 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^(1/2),x, algorithm="maxima")

[Out]

2/13*B*b*x^(13/2) + 2/7*(B*a + A*b)*x^(7/2) + 2*A*a*sqrt(x)

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mupad [B]  time = 2.59, size = 31, normalized size = 0.84 \begin {gather*} \frac {2\,\sqrt {x}\,\left (91\,A\,a+13\,A\,b\,x^3+13\,B\,a\,x^3+7\,B\,b\,x^6\right )}{91} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3))/x^(1/2),x)

[Out]

(2*x^(1/2)*(91*A*a + 13*A*b*x^3 + 13*B*a*x^3 + 7*B*b*x^6))/91

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sympy [A]  time = 2.30, size = 44, normalized size = 1.19 \begin {gather*} 2 A a \sqrt {x} + \frac {2 A b x^{\frac {7}{2}}}{7} + \frac {2 B a x^{\frac {7}{2}}}{7} + \frac {2 B b x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(B*x**3+A)/x**(1/2),x)

[Out]

2*A*a*sqrt(x) + 2*A*b*x**(7/2)/7 + 2*B*a*x**(7/2)/7 + 2*B*b*x**(13/2)/13

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