Optimal. Leaf size=200 \[ \frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}-\frac{105 b e^3}{8 \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140279, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 51, 63, 208} \[ \frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}-\frac{105 b e^3}{8 \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)^{5/2}} \, dx\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}-\frac{(3 e) \int \frac{1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{2 (b d-a e)}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}+\frac{\left (21 e^2\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{8 (b d-a e)^2}\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac{\left (105 e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^3}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac{\left (105 b e^3\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^4}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac{105 b e^3}{8 (b d-a e)^5 \sqrt{d+e x}}-\frac{\left (105 b^2 e^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^5}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac{105 b e^3}{8 (b d-a e)^5 \sqrt{d+e x}}-\frac{\left (105 b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 (b d-a e)^5}\\ &=-\frac{35 e^3}{8 (b d-a e)^4 (d+e x)^{3/2}}-\frac{1}{3 (b d-a e) (a+b x)^3 (d+e x)^{3/2}}+\frac{3 e}{4 (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2}}-\frac{21 e^2}{8 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac{105 b e^3}{8 (b d-a e)^5 \sqrt{d+e x}}+\frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0173553, size = 52, normalized size = 0.26 \[ -\frac{2 e^3 \, _2F_1\left (-\frac{3}{2},4;-\frac{1}{2};-\frac{b (d+e x)}{a e-b d}\right )}{3 (d+e x)^{3/2} (a e-b d)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.212, size = 319, normalized size = 1.6 \begin{align*} -{\frac{2\,{e}^{3}}{3\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+8\,{\frac{{e}^{3}b}{ \left ( ae-bd \right ) ^{5}\sqrt{ex+d}}}+{\frac{41\,{e}^{3}{b}^{4}}{8\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{e}^{4}{b}^{3}a}{3\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{b}^{4}d{e}^{3}}{3\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{e}^{5}{b}^{2}{a}^{2}}{8\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{55\,{e}^{4}{b}^{3}ad}{4\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{55\,{e}^{3}{b}^{4}{d}^{2}}{8\, \left ( ae-bd \right ) ^{5} \left ( bxe+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{105\,{e}^{3}{b}^{2}}{8\, \left ( ae-bd \right ) ^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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 = 2.56335, size = 3717, normalized size = 18.58 \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 = 1.22311, size = 576, normalized size = 2.88 \begin{align*} -\frac{105 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{315 \,{\left (x e + d\right )}^{4} b^{4} e^{3} - 840 \,{\left (x e + d\right )}^{3} b^{4} d e^{3} + 693 \,{\left (x e + d\right )}^{2} b^{4} d^{2} e^{3} - 144 \,{\left (x e + d\right )} b^{4} d^{3} e^{3} - 16 \, b^{4} d^{4} e^{3} + 840 \,{\left (x e + d\right )}^{3} a b^{3} e^{4} - 1386 \,{\left (x e + d\right )}^{2} a b^{3} d e^{4} + 432 \,{\left (x e + d\right )} a b^{3} d^{2} e^{4} + 64 \, a b^{3} d^{3} e^{4} + 693 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{5} - 432 \,{\left (x e + d\right )} a^{2} b^{2} d e^{5} - 96 \, a^{2} b^{2} d^{2} e^{5} + 144 \,{\left (x e + d\right )} a^{3} b e^{6} + 64 \, a^{3} b d e^{6} - 16 \, a^{4} e^{7}}{24 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]