Optimal. Leaf size=144 \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left (a g^2+c f^2\right )}{g \sqrt{f+g x} (e f-d g)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.271231, antiderivative size = 144, normalized size of antiderivative = 1., 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, 1259, 453, 208} \[ -\frac{\sqrt{f+g x} \left (a e^2+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}-\frac{2 \left (a g^2+c f^2\right )}{g \sqrt{f+g x} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 898
Rule 1259
Rule 453
Rule 208
Rubi steps
\begin{align*} \int \frac{a+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, 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}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^2} \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{e (e f-d g)^2 (d+e x)}-\frac{g^3 \operatorname{Subst}\left (\int \frac{\frac{2 e^2 (e f-d g) \left (c f^2+a g^2\right )}{g^5}+\frac{e \left (a e^2 g^2-c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )} \, dx,x,\sqrt{f+g x}\right )}{e^2 (e f-d g)^2}\\ &=-\frac{2 \left (c f^2+a g^2\right )}{g (e f-d g)^2 \sqrt{f+g x}}-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{e (e f-d g)^2 (d+e x)}-\frac{\left (3 a e^2 g+c d (4 e f-d g)\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 )}{e g (e f-d g)^2}\\ &=-\frac{2 \left (c f^2+a g^2\right )}{g (e f-d g)^2 \sqrt{f+g x}}-\frac{\left (c d^2+a e^2\right ) \sqrt{f+g x}}{e (e f-d g)^2 (d+e x)}+\frac{\left (3 a e^2 g+c d (4 e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0878971, size = 118, normalized size = 0.82 \[ -\frac{2 \left (g^2 \left (a e^2+c d^2\right ) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )+2 c d g (e f-d g) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )+c (e f-d g)^2\right )}{e^2 g \sqrt{f+g x} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.245, size = 269, normalized size = 1.9 \begin{align*} -2\,{\frac{ag}{ \left ( dg-ef \right ) ^{2}\sqrt{gx+f}}}-2\,{\frac{c{f}^{2}}{g \left ( dg-ef \right ) ^{2}\sqrt{gx+f}}}-{\frac{aeg}{ \left ( dg-ef \right ) ^{2} \left ( egx+dg \right ) }\sqrt{gx+f}}-{\frac{c{d}^{2}g}{ \left ( dg-ef \right ) ^{2}e \left ( egx+dg \right ) }\sqrt{gx+f}}-3\,{\frac{aeg}{ \left ( dg-ef \right ) ^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+{\frac{c{d}^{2}g}{ \left ( dg-ef \right ) ^{2}e}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-4\,{\frac{cdf}{ \left ( dg-ef \right ) ^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{e\sqrt{gx+f}}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.28051, size = 1817, normalized size = 12.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16049, size = 304, normalized size = 2.11 \begin{align*} \frac{{\left (c d^{2} g - 4 \, c d f e - 3 \, a g e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{{\left (d^{2} g^{2} e - 2 \, d f g e^{2} + f^{2} e^{3}\right )} \sqrt{d g e - f e^{2}}} - \frac{{\left (g x + f\right )} c d^{2} g^{2} + 2 \, c d f^{2} g e + 2 \, a d g^{3} e + 2 \,{\left (g x + f\right )} c f^{2} e^{2} - 2 \, c f^{3} e^{2} + 3 \,{\left (g x + f\right )} a g^{2} e^{2} - 2 \, a f g^{2} e^{2}}{{\left (d^{2} g^{3} e - 2 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )}{\left (\sqrt{g x + f} d g +{\left (g x + f\right )}^{\frac{3}{2}} e - \sqrt{g x + f} f e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]