Optimal. Leaf size=94 \[ -\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt {c d^2-a e^2}}-\frac {\sqrt {d+e x}}{c d (a e+c d x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {626, 47, 63, 208} \begin {gather*} -\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt {c d^2-a e^2}}-\frac {\sqrt {d+e x}}{c d (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 208
Rule 626
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 )^2} \, dx &=\int \frac {\sqrt {d+e x}}{(a e+c d x)^2} \, dx\\ &=-\frac {\sqrt {d+e x}}{c d (a e+c d x)}+\frac {e \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c d}\\ &=-\frac {\sqrt {d+e x}}{c d (a e+c d x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c d}\\ &=-\frac {\sqrt {d+e x}}{c d (a e+c d x)}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt {c d^2-a e^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 93, normalized size = 0.99 \begin {gather*} \frac {e \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {a e^2-c d^2}}\right )}{c^{3/2} d^{3/2} \sqrt {a e^2-c d^2}}-\frac {\sqrt {d+e x}}{a c d e+c^2 d^2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.26, size = 121, normalized size = 1.29 \begin {gather*} \frac {e \sqrt {d+e x}}{c d \left (-a e^2+c d^2-c d (d+e x)\right )}-\frac {e \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e^2-c d^2}}{c d^2-a e^2}\right )}{c^{3/2} d^{3/2} \sqrt {a e^2-c d^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 309, normalized size = 3.29 \begin {gather*} \left [\frac {\sqrt {c^{2} d^{3} - a c d e^{2}} {\left (c d e x + a e^{2}\right )} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right ) - 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} + {\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x\right )}}, \frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d e x + a e^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right ) - {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} + {\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [A] time = 0.06, size = 84, normalized size = 0.89 \begin {gather*} \frac {e \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c d}-\frac {\sqrt {e x +d}\, e}{\left (c d e x +a \,e^{2}\right ) c d} \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 = 0.65, size = 81, normalized size = 0.86 \begin {gather*} \frac {e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}}{\sqrt {a\,e^2-c\,d^2}}\right )}{c^{3/2}\,d^{3/2}\,\sqrt {a\,e^2-c\,d^2}}-\frac {e\,\sqrt {d+e\,x}}{x\,c^2\,d^2\,e+a\,c\,d\,e^2} \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]
________________________________________________________________________________________