Optimal. Leaf size=67 \[ -\frac {2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt {a+b x^3}}-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {453, 264} \begin {gather*} -\frac {2 (e x)^{3/2} (2 A b-a B)}{3 a^2 e^4 \sqrt {a+b x^3}}-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^3}{(e x)^{5/2} \left (a+b x^3\right )^{3/2}} \, dx &=-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}}-\frac {(2 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^3\right )^{3/2}} \, dx}{a e^3}\\ &=-\frac {2 A}{3 a e (e x)^{3/2} \sqrt {a+b x^3}}-\frac {2 (2 A b-a B) (e x)^{3/2}}{3 a^2 e^4 \sqrt {a+b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 45, normalized size = 0.67 \begin {gather*} \frac {x \left (-2 a A+2 a B x^3-4 A b x^3\right )}{3 a^2 (e x)^{5/2} \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.04, size = 71, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-a A e^3+a B e^3 x^3-2 A b e^3 x^3\right )}{3 a^2 e (e x)^{3/2} \left (a e^3+b e^3 x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 57, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left ({\left (B a - 2 \, A b\right )} x^{3} - A a\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{3 \, {\left (a^{2} b e^{3} x^{5} + a^{3} e^{3} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 39, normalized size = 0.58 \begin {gather*} -\frac {2 \left (2 A \,x^{3} b -B a \,x^{3}+A a \right ) x}{3 \sqrt {b \,x^{3}+a}\, \left (e x \right )^{\frac {5}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.75, size = 70, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {2\,A}{3\,a\,b\,e^2}+\frac {x^3\,\left (4\,A\,b-2\,B\,a\right )}{3\,a^2\,b\,e^2}\right )\,\sqrt {b\,x^3+a}}{x^4\,\sqrt {e\,x}+\frac {a\,x\,\sqrt {e\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________