Optimal. Leaf size=171 \[ \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 650} \begin {gather*} \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x) \left (c d^2-a e^2\right )^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 (d+e x)^3 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {(4 c d) \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {\left (8 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{15 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^2 (d+e x)^2}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 \left (c d^2-a e^2\right )^3 (d+e x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (3 a^2 e^4-2 a c d e^2 (5 d+2 e x)+c^2 d^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x)^3 \left (c d^2-a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.06, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.80, size = 279, normalized size = 1.63 \begin {gather*} \frac {2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 4 \, {\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \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: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 146, normalized size = 0.85 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (8 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x +20 c^{2} d^{3} e x +3 a^{2} e^{4}-10 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right )}{15 \left (e x +d \right )^{2} \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.78, size = 110, normalized size = 0.64 \begin {gather*} -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^2\,e^4-10\,a\,c\,d^2\,e^2-4\,a\,c\,d\,e^3\,x+15\,c^2\,d^4+20\,c^2\,d^3\,e\,x+8\,c^2\,d^2\,e^2\,x^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________