Optimal. Leaf size=245 \[ \frac{\sqrt{d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}+\frac{c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}-\frac{\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.377742, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {736, 822, 826, 1166, 208} \[ \frac{\sqrt{d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}+\frac{c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}-\frac{\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 736
Rule 822
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\int \frac{-6 c d+\frac{b e}{2}-5 c e x}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{4} (c d-b e) \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )-\frac{1}{4} c e \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} e (c d-b e) \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )+\frac{1}{4} c d e \left (24 c^2 d^2-24 b c d e+b^2 e^2\right )-\frac{1}{4} c e \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4 d (c d-b e)}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac{\left (c \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 d}-\frac{\left (c^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 (c d-b e)}\\ &=-\frac{(b+2 c x) \sqrt{d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac{\left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{3/2}}+\frac{c^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.975623, size = 361, normalized size = 1.47 \[ \frac{-\frac{2 c (d+e x)^{3/2} \left (-b^2 e^2-9 b c d e+12 c^2 d^2\right )}{b^2 d (b e-c d)}+\frac{(b+c x) \left (2 b c^{5/2} (d+e x)^{3/2} \left (10 b^2 c d e^2+b^3 e^3-36 b c^2 d^2 e+24 c^3 d^3\right )+(b+c x) \left (2 c^{3/2} (c d-b e)^2 \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \left (\sqrt{d+e x}-\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )-2 c^3 d^2 \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \left (\sqrt{c} \sqrt{d+e x}-\sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )\right )\right )}{b^4 c^{3/2} d (c d-b e)^2}+\frac{2 (d+e x)^{3/2} (b e+8 c d)}{b d x}-\frac{4 (d+e x)^{3/2}}{x^2}}{8 b d (b+c x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.234, size = 436, normalized size = 1.8 \begin{align*}{\frac{11\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-3\,{\frac{e{c}^{4} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }}+{\frac{13\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{3}\sqrt{ex+d}d}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{35\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( be-cd \right ) }\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-21\,{\frac{e{c}^{3}d}{{b}^{4} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+12\,{\frac{{c}^{4}{d}^{2}}{{b}^{5} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{4\,{b}^{3}{x}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}}}-3\,{\frac{\sqrt{ex+d}cd}{e{b}^{4}{x}^{2}}}-{\frac{1}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}+{\frac{{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+3\,{\frac{ce}{{b}^{4}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{\sqrt{d}{c}^{2}}{{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \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 = 4.48442, size = 4645, normalized size = 18.96 \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 [B] time = 1.27475, size = 686, normalized size = 2.8 \begin{align*} -\frac{{\left (48 \, c^{4} d^{2} - 84 \, b c^{3} d e + 35 \, b^{2} c^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \,{\left (b^{5} c d - b^{6} e\right )} \sqrt{-c^{2} d + b c e}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{4} d^{2} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{4} d^{3} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{4} e - 24 \, \sqrt{x e + d} c^{4} d^{5} e - 24 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{3} d e^{2} + 108 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{3} d^{2} e^{2} - 144 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{3} e^{2} + 60 \, \sqrt{x e + d} b c^{3} d^{4} e^{2} +{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{2} d e^{3} + 85 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{2} e^{3} - 46 \, \sqrt{x e + d} b^{2} c^{2} d^{3} e^{3} + 2 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c e^{4} - 13 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c d e^{4} + 9 \, \sqrt{x e + d} b^{3} c d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{5} + \sqrt{x e + d} b^{4} d e^{5}}{4 \,{\left (b^{4} c d^{2} - b^{5} d e\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac{{\left (48 \, c^{2} d^{2} - 12 \, b c d e - b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]