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

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(19/2))/19

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Rubi [A]  time = 0.0291588, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ 2 a^2 A \sqrt{x}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{7} a x^{7/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(7/2))/7 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(19/2))/19

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 )^2 \left (A+B x^3\right )}{\sqrt{x}} \, dx &=\int \left (\frac{a^2 A}{\sqrt{x}}+a (2 A b+a B) x^{5/2}+b (A b+2 a B) x^{11/2}+b^2 B x^{17/2}\right ) \, dx\\ &=2 a^2 A \sqrt{x}+\frac{2}{7} a (2 A b+a B) x^{7/2}+\frac{2}{13} b (A b+2 a B) x^{13/2}+\frac{2}{19} b^2 B x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0292543, size = 53, normalized size = 0.87 \[ \frac{2 \sqrt{x} \left (1729 a^2 A+133 b x^6 (2 a B+A b)+247 a x^3 (a B+2 A b)+91 b^2 B x^9\right )}{1729} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(1729*a^2*A + 247*a*(2*A*b + a*B)*x^3 + 133*b*(A*b + 2*a*B)*x^6 + 91*b^2*B*x^9))/1729

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Maple [A]  time = 0.007, size = 56, normalized size = 0.9 \begin{align*}{\frac{182\,B{b}^{2}{x}^{9}+266\,A{b}^{2}{x}^{6}+532\,B{x}^{6}ab+988\,aAb{x}^{3}+494\,B{x}^{3}{a}^{2}+3458\,{a}^{2}A}{1729}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1729*x^(1/2)*(91*B*b^2*x^9+133*A*b^2*x^6+266*B*a*b*x^6+494*A*a*b*x^3+247*B*a^2*x^3+1729*A*a^2)

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Maxima [A]  time = 0.948468, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{7} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(1/2),x, algorithm="maxima")

[Out]

2/19*B*b^2*x^(19/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/7*(B*a^2 + 2*A*a*b)*x^(7/2) + 2*A*a^2*sqrt(x)

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Fricas [A]  time = 1.6437, size = 135, normalized size = 2.21 \begin{align*} \frac{2}{1729} \,{\left (91 \, B b^{2} x^{9} + 133 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 247 \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + 1729 \, A a^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(1/2),x, algorithm="fricas")

[Out]

2/1729*(91*B*b^2*x^9 + 133*(2*B*a*b + A*b^2)*x^6 + 247*(B*a^2 + 2*A*a*b)*x^3 + 1729*A*a^2)*sqrt(x)

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Sympy [A]  time = 7.22549, size = 78, normalized size = 1.28 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{7}{2}}}{7} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(7/2)/7 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x**(7/2)/7 + 4*B*a*b*x**(13/2)/13 + 2
*B*b**2*x**(19/2)/19

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Giac [A]  time = 1.10846, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a b x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(B*x^3+A)/x^(1/2),x, algorithm="giac")

[Out]

2/19*B*b^2*x^(19/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/7*B*a^2*x^(7/2) + 4/7*A*a*b*x^(7/2) + 2*A*
a^2*sqrt(x)