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

Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {2}{17} b^2 x^{17/2} (3 a B+A b)+\frac {6}{11} a b x^{11/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]

________________________________________________________________________________________

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} \begin {gather*} \frac {2}{5} a^2 x^{5/2} (a B+3 A b)-\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{17} b^2 x^{17/2} (3 a B+A b)+\frac {6}{11} a b x^{11/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(5/2))/5 + (6*a*b*(A*b + a*B)*x^(11/2))/11 + (2*b^2*(A*b + 3*a*B)*
x^(17/2))/17 + (2*b^3*B*x^(23/2))/23

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 )}{x^{3/2}} \, dx &=\int \left (\frac {a^3 A}{x^{3/2}}+a^2 (3 A b+a B) x^{3/2}+3 a b (A b+a B) x^{9/2}+b^2 (A b+3 a B) x^{15/2}+b^3 B x^{21/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {6}{11} a b (A b+a B) x^{11/2}+\frac {2}{17} b^2 (A b+3 a B) x^{17/2}+\frac {2}{23} b^3 B x^{23/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 81, normalized size = 0.98 \begin {gather*} \frac {-8602 a^3 \left (5 A-B x^3\right )+2346 a^2 b x^3 \left (11 A+5 B x^3\right )+690 a b^2 x^6 \left (17 A+11 B x^3\right )+110 b^3 x^9 \left (23 A+17 B x^3\right )}{21505 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8602*a^3*(5*A - B*x^3) + 2346*a^2*b*x^3*(11*A + 5*B*x^3) + 690*a*b^2*x^6*(17*A + 11*B*x^3) + 110*b^3*x^9*(23
*A + 17*B*x^3))/(21505*Sqrt[x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 83, normalized size = 1.00 \begin {gather*} \frac {2 \left (-21505 a^3 A+4301 a^3 B x^3+12903 a^2 A b x^3+5865 a^2 b B x^6+5865 a A b^2 x^6+3795 a b^2 B x^9+1265 A b^3 x^9+935 b^3 B x^{12}\right )}{21505 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-21505*a^3*A + 12903*a^2*A*b*x^3 + 4301*a^3*B*x^3 + 5865*a*A*b^2*x^6 + 5865*a^2*b*B*x^6 + 1265*A*b^3*x^9 +
3795*a*b^2*B*x^9 + 935*b^3*B*x^12))/(21505*Sqrt[x])

________________________________________________________________________________________

fricas [A]  time = 1.00, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (935 \, B b^{3} x^{12} + 1265 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 5865 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} - 21505 \, A a^{3} + 4301 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )}}{21505 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21505*(935*B*b^3*x^12 + 1265*(3*B*a*b^2 + A*b^3)*x^9 + 5865*(B*a^2*b + A*a*b^2)*x^6 - 21505*A*a^3 + 4301*(B*
a^3 + 3*A*a^2*b)*x^3)/sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {6}{17} \, B a b^{2} x^{\frac {17}{2}} + \frac {2}{17} \, A b^{3} x^{\frac {17}{2}} + \frac {6}{11} \, B a^{2} b x^{\frac {11}{2}} + \frac {6}{11} \, A a b^{2} x^{\frac {11}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*B*b^3*x^(23/2) + 6/17*B*a*b^2*x^(17/2) + 2/17*A*b^3*x^(17/2) + 6/11*B*a^2*b*x^(11/2) + 6/11*A*a*b^2*x^(11
/2) + 2/5*B*a^3*x^(5/2) + 6/5*A*a^2*b*x^(5/2) - 2*A*a^3/sqrt(x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 80, normalized size = 0.96 \begin {gather*} -\frac {2 \left (-935 B \,b^{3} x^{12}-1265 x^{9} A \,b^{3}-3795 x^{9} B a \,b^{2}-5865 x^{6} A a \,b^{2}-5865 x^{6} B \,a^{2} b -12903 x^{3} A \,a^{2} b -4301 B \,a^{3} x^{3}+21505 A \,a^{3}\right )}{21505 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21505*(-935*B*b^3*x^12-1265*A*b^3*x^9-3795*B*a*b^2*x^9-5865*A*a*b^2*x^6-5865*B*a^2*b*x^6-12903*A*a^2*b*x^3-
4301*B*a^3*x^3+21505*A*a^3)/x^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {2}{17} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {17}{2}} + \frac {6}{11} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {11}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*B*b^3*x^(23/2) + 2/17*(3*B*a*b^2 + A*b^3)*x^(17/2) + 6/11*(B*a^2*b + A*a*b^2)*x^(11/2) - 2*A*a^3/sqrt(x)
+ 2/5*(B*a^3 + 3*A*a^2*b)*x^(5/2)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 69, normalized size = 0.83 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{17/2}\,\left (\frac {2\,A\,b^3}{17}+\frac {6\,B\,a\,b^2}{17}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{23/2}}{23}+\frac {6\,a\,b\,x^{11/2}\,\left (A\,b+B\,a\right )}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(5/2)*((2*B*a^3)/5 + (6*A*a^2*b)/5) + x^(17/2)*((2*A*b^3)/17 + (6*B*a*b^2)/17) - (2*A*a^3)/x^(1/2) + (2*B*b^
3*x^(23/2))/23 + (6*a*b*x^(11/2)*(A*b + B*a))/11

________________________________________________________________________________________

sympy [A]  time = 19.49, size = 112, normalized size = 1.35 \begin {gather*} - \frac {2 A a^{3}}{\sqrt {x}} + \frac {6 A a^{2} b x^{\frac {5}{2}}}{5} + \frac {6 A a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 A b^{3} x^{\frac {17}{2}}}{17} + \frac {2 B a^{3} x^{\frac {5}{2}}}{5} + \frac {6 B a^{2} b x^{\frac {11}{2}}}{11} + \frac {6 B a b^{2} x^{\frac {17}{2}}}{17} + \frac {2 B b^{3} x^{\frac {23}{2}}}{23} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**3/sqrt(x) + 6*A*a**2*b*x**(5/2)/5 + 6*A*a*b**2*x**(11/2)/11 + 2*A*b**3*x**(17/2)/17 + 2*B*a**3*x**(5/2
)/5 + 6*B*a**2*b*x**(11/2)/11 + 6*B*a*b**2*x**(17/2)/17 + 2*B*b**3*x**(23/2)/23

________________________________________________________________________________________