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

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

[Out]

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

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Rubi [A]  time = 0.0148383, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{7} x^{7/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{13} b B x^{13/2} \]

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*}

Mathematica [A]  time = 0.0134174, size = 33, normalized size = 0.89 \[ \frac{2}{91} \sqrt{x} \left (13 x^3 (a B+A b)+91 a A+7 b B x^6\right ) \]

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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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \begin{align*}{\frac{14\,bB{x}^{6}+26\,A{x}^{3}b+26\,B{x}^{3}a+182\,Aa}{91}\sqrt{x}} \end{align*}

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 = 0.947247, size = 36, normalized size = 0.97 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.70982, size = 76, normalized size = 2.05 \begin{align*} \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{align*}

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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Sympy [A]  time = 2.07037, size = 44, normalized size = 1.19 \begin{align*} 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{align*}

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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Giac [A]  time = 1.10385, size = 39, normalized size = 1.05 \begin{align*} \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{align*}

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)