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

Optimal. Leaf size=83 \[ \frac{2}{7} a^2 x^{7/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(
19/2))/19 + (2*b^3*B*x^(25/2))/25

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Rubi [A]  time = 0.0427669, antiderivative size = 83, 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} \[ \frac{2}{7} a^2 x^{7/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(
19/2))/19 + (2*b^3*B*x^(25/2))/25

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 )^3 \left (A+B x^3\right )}{\sqrt{x}} \, dx &=\int \left (\frac{a^3 A}{\sqrt{x}}+a^2 (3 A b+a B) x^{5/2}+3 a b (A b+a B) x^{11/2}+b^2 (A b+3 a B) x^{17/2}+b^3 B x^{23/2}\right ) \, dx\\ &=2 a^3 A \sqrt{x}+\frac{2}{7} a^2 (3 A b+a B) x^{7/2}+\frac{6}{13} a b (A b+a B) x^{13/2}+\frac{2}{19} b^2 (A b+3 a B) x^{19/2}+\frac{2}{25} b^3 B x^{25/2}\\ \end{align*}

Mathematica [A]  time = 0.038728, size = 83, normalized size = 1. \[ \frac{2}{7} a^2 x^{7/2} (a B+3 A b)+2 a^3 A \sqrt{x}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^3*A*Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(7/2))/7 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(
19/2))/19 + (2*b^3*B*x^(25/2))/25

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Maple [A]  time = 0.009, size = 80, normalized size = 1. \begin{align*}{\frac{3458\,{b}^{3}B{x}^{12}+4550\,{x}^{9}{b}^{3}A+13650\,{x}^{9}a{b}^{2}B+19950\,{x}^{6}a{b}^{2}A+19950\,{x}^{6}{a}^{2}bB+37050\,{x}^{3}A{a}^{2}b+12350\,{x}^{3}B{a}^{3}+86450\,{a}^{3}A}{43225}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/43225*x^(1/2)*(1729*B*b^3*x^12+2275*A*b^3*x^9+6825*B*a*b^2*x^9+9975*A*a*b^2*x^6+9975*B*a^2*b*x^6+18525*A*a^2
*b*x^3+6175*B*a^3*x^3+43225*A*a^3)

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Maxima [A]  time = 0.941228, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{25} \, B b^{3} x^{\frac{25}{2}} + \frac{2}{19} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{19}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{7} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*B*b^3*x^(25/2) + 2/19*(3*B*a*b^2 + A*b^3)*x^(19/2) + 6/13*(B*a^2*b + A*a*b^2)*x^(13/2) + 2*A*a^3*sqrt(x)
+ 2/7*(B*a^3 + 3*A*a^2*b)*x^(7/2)

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Fricas [A]  time = 1.67248, size = 192, normalized size = 2.31 \begin{align*} \frac{2}{43225} \,{\left (1729 \, B b^{3} x^{12} + 2275 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 9975 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 43225 \, A a^{3} + 6175 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/43225*(1729*B*b^3*x^12 + 2275*(3*B*a*b^2 + A*b^3)*x^9 + 9975*(B*a^2*b + A*a*b^2)*x^6 + 43225*A*a^3 + 6175*(B
*a^3 + 3*A*a^2*b)*x^3)*sqrt(x)

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Sympy [A]  time = 18.321, size = 112, normalized size = 1.35 \begin{align*} 2 A a^{3} \sqrt{x} + \frac{6 A a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{7}{2}}}{7} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{19}{2}}}{19} + \frac{2 B b^{3} x^{\frac{25}{2}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*sqrt(x) + 6*A*a**2*b*x**(7/2)/7 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b**3*x**(19/2)/19 + 2*B*a**3*x**(7/2)
/7 + 6*B*a**2*b*x**(13/2)/13 + 6*B*a*b**2*x**(19/2)/19 + 2*B*b**3*x**(25/2)/25

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Giac [A]  time = 1.13561, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{25} \, B b^{3} x^{\frac{25}{2}} + \frac{6}{19} \, B a b^{2} x^{\frac{19}{2}} + \frac{2}{19} \, A b^{3} x^{\frac{19}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{6}{7} \, A a^{2} b x^{\frac{7}{2}} + 2 \, A a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/25*B*b^3*x^(25/2) + 6/19*B*a*b^2*x^(19/2) + 2/19*A*b^3*x^(19/2) + 6/13*B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13
/2) + 2/7*B*a^3*x^(7/2) + 6/7*A*a^2*b*x^(7/2) + 2*A*a^3*sqrt(x)