Optimal. Leaf size=331 \[ -\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2}{24 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac{7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.246489, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \[ -\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2}{24 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac{7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 672
Rule 666
Rule 660
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(7 c d) \int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{6 \left (c d^2-a e^2\right )}\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (35 c^2 d^2\right ) \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{24 \left (c d^2-a e^2\right )^2}\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (35 c^3 d^3\right ) \int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{16 \left (c d^2-a e^2\right )^3}\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (35 c^3 d^3 e\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 \left (c d^2-a e^2\right )^4}\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (35 c^3 d^3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{8 \left (c d^2-a e^2\right )^4}\\ &=\frac{1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d^2-a e^2} \sqrt{d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0259689, size = 81, normalized size = 0.24 \[ -\frac{2 c^3 d^3 \sqrt{d+e x} \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{\left (c d^2-a e^2\right )^4 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.251, size = 559, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 24\,cdx+24\,ae \right ) \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{3}{c}^{3}{d}^{3}{e}^{4}+315\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{3}{d}^{4}{e}^{3}+315\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{3}{d}^{5}{e}^{2}-105\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{3}{c}^{3}{d}^{3}{e}^{3}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{c}^{3}{d}^{6}e-35\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-280\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{3}{d}^{4}{e}^{2}+14\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{a}^{2}cd{e}^{5}-98\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xa{c}^{2}{d}^{3}{e}^{3}-231\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{3}{d}^{5}e-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{3}{e}^{6}+38\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}c{d}^{2}{e}^{4}-87\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{c}^{2}{d}^{4}{e}^{2}-48\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.46908, size = 3243, normalized size = 9.8 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]