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

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

[Out]

2/7*a^2*(3*A*b+B*a)*x^(7/2)+6/13*a*b*(A*b+B*a)*x^(13/2)+2/19*b^2*(A*b+3*B*a)*x^(19/2)+2/25*b^3*B*x^(25/2)+2*a^
3*A*x^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.06, size = 83, normalized size = 1.00 \[ 2 a^3 A \sqrt {x}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {6}{13} a b x^{13/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2} \]

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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fricas [A]  time = 1.45, size = 75, normalized size = 0.90 \[ \frac {2}{43225} \, {\left (1729 \, B b^{3} x^{12} + 2275 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 9975 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + 43225 \, A a^{3} + 6175 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \]

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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giac [A]  time = 0.16, size = 77, normalized size = 0.93 \[ \frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {6}{19} \, B a b^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{3} x^{\frac {19}{2}} + \frac {6}{13} \, B a^{2} b x^{\frac {13}{2}} + \frac {6}{13} \, A a b^{2} x^{\frac {13}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + 2 \, A a^{3} \sqrt {x} \]

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)

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maple [A]  time = 0.04, size = 80, normalized size = 0.96 \[ \frac {2 \left (1729 B \,b^{3} x^{12}+2275 x^{9} A \,b^{3}+6825 x^{9} B a \,b^{2}+9975 x^{6} A a \,b^{2}+9975 x^{6} B \,a^{2} b +18525 x^{3} A \,a^{2} b +6175 B \,a^{3} x^{3}+43225 A \,a^{3}\right ) \sqrt {x}}{43225} \]

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 = 0.56, size = 73, normalized size = 0.88 \[ \frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{19} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {19}{2}} + \frac {6}{13} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {13}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \]

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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mupad [B]  time = 0.03, size = 69, normalized size = 0.83 \[ x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{19/2}\,\left (\frac {2\,A\,b^3}{19}+\frac {6\,B\,a\,b^2}{19}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{25/2}}{25}+\frac {6\,a\,b\,x^{13/2}\,\left (A\,b+B\,a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(7/2)*((2*B*a^3)/7 + (6*A*a^2*b)/7) + x^(19/2)*((2*A*b^3)/19 + (6*B*a*b^2)/19) + 2*A*a^3*x^(1/2) + (2*B*b^3*
x^(25/2))/25 + (6*a*b*x^(13/2)*(A*b + B*a))/13

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sympy [A]  time = 23.64, size = 112, normalized size = 1.35 \[ 2 A a^{3} \sqrt {x} + \frac {6 A a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 A a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 A b^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {6 B a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 B a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {25}{2}}}{25} \]

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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