Optimal. Leaf size=119 \[ -\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {2 (b c-a d)}{(d e-c f)^2 \sqrt {e+f x}}+\frac {2 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 214}
\begin {gather*} -\frac {2 (b c-a d)}{\sqrt {e+f x} (d e-c f)^2}-\frac {2 (b e-a f)}{3 f (e+f x)^{3/2} (d e-c f)}+\frac {2 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps
\begin {align*} \int \frac {a+b x}{(c+d x) (e+f x)^{5/2}} \, dx &=-\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {(b c-a d) \int \frac {1}{(c+d x) (e+f x)^{3/2}} \, dx}{d e-c f}\\ &=-\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {2 (b c-a d)}{(d e-c f)^2 \sqrt {e+f x}}-\frac {(d (b c-a d)) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{(d e-c f)^2}\\ &=-\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {2 (b c-a d)}{(d e-c f)^2 \sqrt {e+f x}}-\frac {(2 d (b c-a d)) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{f (d e-c f)^2}\\ &=-\frac {2 (b e-a f)}{3 f (d e-c f) (e+f x)^{3/2}}-\frac {2 (b c-a d)}{(d e-c f)^2 \sqrt {e+f x}}+\frac {2 \sqrt {d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.27, size = 117, normalized size = 0.98 \begin {gather*} -\frac {2 \left (b d e^2+b c f (2 e+3 f x)+a f (-4 d e+c f-3 d f x)\right )}{3 f (d e-c f)^2 (e+f x)^{3/2}}+\frac {2 \sqrt {d} (-b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 116, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {2 d f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f}\) | \(116\) |
default | \(\frac {\frac {2 d f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{3 \left (c f -d e \right ) \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}}{f}\) | \(116\) |
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]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs.
\(2 (111) = 222\).
time = 1.30, size = 515, normalized size = 4.33 \begin {gather*} \left [-\frac {3 \, {\left ({\left (b c - a d\right )} f^{3} x^{2} + 2 \, {\left (b c - a d\right )} f^{2} x e + {\left (b c - a d\right )} f e^{2}\right )} \sqrt {-\frac {d}{c f - d e}} \log \left (\frac {d f x - c f + 2 \, {\left (c f - d e\right )} \sqrt {f x + e} \sqrt {-\frac {d}{c f - d e}} + 2 \, d e}{d x + c}\right ) + 2 \, {\left (a c f^{2} + 3 \, {\left (b c - a d\right )} f^{2} x + b d e^{2} + 2 \, {\left (b c - 2 \, a d\right )} f e\right )} \sqrt {f x + e}}{3 \, {\left (c^{2} f^{5} x^{2} + d^{2} f e^{4} + 2 \, {\left (d^{2} f^{2} x - c d f^{2}\right )} e^{3} + {\left (d^{2} f^{3} x^{2} - 4 \, c d f^{3} x + c^{2} f^{3}\right )} e^{2} - 2 \, {\left (c d f^{4} x^{2} - c^{2} f^{4} x\right )} e\right )}}, -\frac {2 \, {\left (3 \, {\left ({\left (b c - a d\right )} f^{3} x^{2} + 2 \, {\left (b c - a d\right )} f^{2} x e + {\left (b c - a d\right )} f e^{2}\right )} \sqrt {\frac {d}{c f - d e}} \arctan \left (-\frac {{\left (c f - d e\right )} \sqrt {f x + e} \sqrt {\frac {d}{c f - d e}}}{d f x + d e}\right ) + {\left (a c f^{2} + 3 \, {\left (b c - a d\right )} f^{2} x + b d e^{2} + 2 \, {\left (b c - 2 \, a d\right )} f e\right )} \sqrt {f x + e}\right )}}{3 \, {\left (c^{2} f^{5} x^{2} + d^{2} f e^{4} + 2 \, {\left (d^{2} f^{2} x - c d f^{2}\right )} e^{3} + {\left (d^{2} f^{3} x^{2} - 4 \, c d f^{3} x + c^{2} f^{3}\right )} e^{2} - 2 \, {\left (c d f^{4} x^{2} - c^{2} f^{4} x\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 26.58, size = 105, normalized size = 0.88 \begin {gather*} \frac {2 \left (a d - b c\right )}{\sqrt {e + f x} \left (c f - d e\right )^{2}} + \frac {2 \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{2}} - \frac {2 \left (a f - b e\right )}{3 f \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.68, size = 160, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{{\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \sqrt {c d f - d^{2} e}} - \frac {2 \, {\left (3 \, {\left (f x + e\right )} b c f - 3 \, {\left (f x + e\right )} a d f + a c f^{2} - b c f e - a d f e + b d e^{2}\right )}}{3 \, {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} {\left (f x + e\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.19, size = 128, normalized size = 1.08 \begin {gather*} \frac {2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^{5/2}}\right )\,\left (a\,d-b\,c\right )}{{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {2\,\left (a\,f-b\,e\right )}{3\,\left (c\,f-d\,e\right )}-\frac {2\,\left (e+f\,x\right )\,\left (a\,d\,f-b\,c\,f\right )}{{\left (c\,f-d\,e\right )}^2}}{f\,{\left (e+f\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________