Optimal. Leaf size=105 \[ \frac {2 (d+e x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \begin {gather*} \frac {2 (d+e x)^{3/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 656
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2 (d+e x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d e}\\ &=-\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{3/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 51, normalized size = 0.49 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (c d (d-e x)-2 a e^2\right )}{c^2 d^2 \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.93, size = 65, normalized size = 0.62 \begin {gather*} \frac {2 (d+e x)^{3/2} (a e+c d x) \left (e (a e+c d x)+a e^2-c d^2\right )}{c^2 d^2 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 98, normalized size = 0.93 \begin {gather*} \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x - c d^{2} + 2 \, a e^{2}\right )} \sqrt {e x + d}}{c^{3} d^{3} e x^{2} + a c^{2} d^{3} e + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 68, normalized size = 0.65 \begin {gather*} \frac {2 \left (c d x +a e \right ) \left (c d e x +2 a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 36, normalized size = 0.34 \begin {gather*} \frac {2 \, {\left (c d e x - c d^{2} + 2 \, a e^{2}\right )}}{\sqrt {c d x + a e} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.96, size = 116, normalized size = 1.10 \begin {gather*} \frac {\left (\frac {2\,x\,\sqrt {d+e\,x}}{c^2\,d^2}+\frac {\left (4\,a\,e^2-2\,c\,d^2\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (c^3\,d^4+a\,c^2\,d^2\,e^2\right )}{c^3\,d^3\,e}} \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]
________________________________________________________________________________________