Optimal. Leaf size=178 \[ -\frac {\sqrt {f+g x} \left (a+\frac {c d^2}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}-\frac {\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}+\frac {\sqrt {f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{4 e^2 (d+e x) (e f-d g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {898, 1157, 385, 208} \[ -\frac {\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}-\frac {\sqrt {f+g x} \left (a+\frac {c d^2}{e^2}\right )}{2 (d+e x)^2 (e f-d g)}+\frac {\sqrt {f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{4 e^2 (d+e x) (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 385
Rule 898
Rule 1157
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\operatorname {Subst}\left (\int \frac {-3 a+\frac {c d^2}{e^2}-\frac {4 c f^2}{g^2}+\frac {4 c (e f-d g) x^2}{e g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 (e f-d g)}\\ &=-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 g (e f-d g)^2}\\ &=-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.82, size = 207, normalized size = 1.16 \[ \frac {2 \left (\frac {\sqrt {e} g^2 \sqrt {f+g x} \left (a e^2+c d^2\right ) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {e (f+g x)}{e f-d g}\right )}{(d g-e f)^3}-\frac {c d \left (\sqrt {e} \sqrt {f+g x} (d g-e f)+g (d+e x) \sqrt {d g-e f} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {d g-e f}}\right )\right )}{(d+e x) (e f-d g)^2}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}\right )}{e^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.07, size = 896, normalized size = 5.03 \[ \left [\frac {{\left (8 \, c d^{2} e^{2} f^{2} - 8 \, c d^{3} e f g + 3 \, {\left (c d^{4} + a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 8 \, c d e^{3} f g + 3 \, {\left (c d^{2} e^{2} + a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 8 \, c d^{2} e^{2} f g + 3 \, {\left (c d^{3} e + a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) + 2 \, {\left (2 \, {\left (3 \, c d^{2} e^{3} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (8 \, c d e^{4} f^{2} - {\left (13 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{8 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}, \frac {{\left (8 \, c d^{2} e^{2} f^{2} - 8 \, c d^{3} e f g + 3 \, {\left (c d^{4} + a d^{2} e^{2}\right )} g^{2} + {\left (8 \, c e^{4} f^{2} - 8 \, c d e^{3} f g + 3 \, {\left (c d^{2} e^{2} + a e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (8 \, c d e^{3} f^{2} - 8 \, c d^{2} e^{2} f g + 3 \, {\left (c d^{3} e + a d e^{3}\right )} g^{2}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, {\left (3 \, c d^{2} e^{3} - a e^{5}\right )} f^{2} - {\left (9 \, c d^{3} e^{2} - 7 \, a d e^{4}\right )} f g + {\left (3 \, c d^{4} e - 5 \, a d^{2} e^{3}\right )} g^{2} + {\left (8 \, c d e^{4} f^{2} - {\left (13 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} f g + {\left (5 \, c d^{3} e^{2} - 3 \, a d e^{4}\right )} g^{2}\right )} x\right )} \sqrt {g x + f}}{4 \, {\left (d^{2} e^{6} f^{3} - 3 \, d^{3} e^{5} f^{2} g + 3 \, d^{4} e^{4} f g^{2} - d^{5} e^{3} g^{3} + {\left (e^{8} f^{3} - 3 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} - d^{3} e^{5} g^{3}\right )} x^{2} + 2 \, {\left (d e^{7} f^{3} - 3 \, d^{2} e^{6} f^{2} g + 3 \, d^{3} e^{5} f g^{2} - d^{4} e^{4} g^{3}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 278, normalized size = 1.56 \[ \frac {{\left (3 \, c d^{2} g^{2} - 8 \, c d f g e + 8 \, c f^{2} e^{2} + 3 \, a g^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} \sqrt {d g e - f e^{2}}} - \frac {3 \, \sqrt {g x + f} c d^{3} g^{3} + 5 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{2} e - 11 \, \sqrt {g x + f} c d^{2} f g^{2} e - 8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f g e^{2} + 8 \, \sqrt {g x + f} c d f^{2} g e^{2} - 5 \, \sqrt {g x + f} a d g^{3} e^{2} - 3 \, {\left (g x + f\right )}^{\frac {3}{2}} a g^{2} e^{3} + 5 \, \sqrt {g x + f} a f g^{2} e^{3}}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 384, normalized size = 2.16 \[ \frac {3 a \,g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}+\frac {3 c \,d^{2} g^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}\, e^{2}}-\frac {2 c d f g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}\, e}+\frac {2 c \,f^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \sqrt {\left (d g -e f \right ) e}}+\frac {\frac {\left (3 a \,e^{2} g -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}} g}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e}+\frac {\left (5 a \,e^{2} g -3 c \,d^{2} g +8 c d e f \right ) \sqrt {g x +f}\, g}{4 \left (d g -e f \right ) e^{2}}}{\left (d g -e f +\left (g x +f \right ) e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.91, size = 224, normalized size = 1.26 \[ \frac {\frac {\sqrt {f+g\,x}\,\left (-3\,c\,d^2\,g^2+8\,c\,f\,d\,e\,g+5\,a\,e^2\,g^2\right )}{4\,e^2\,\left (d\,g-e\,f\right )}+\frac {{\left (f+g\,x\right )}^{3/2}\,\left (-5\,c\,d^2\,g^2+8\,c\,f\,d\,e\,g+3\,a\,e^2\,g^2\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^2-\left (f+g\,x\right )\,\left (2\,e^2\,f-2\,d\,e\,g\right )+d^2\,g^2+e^2\,f^2-2\,d\,e\,f\,g}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (3\,c\,d^2\,g^2-8\,c\,d\,e\,f\,g+8\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{4\,e^{5/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________