Optimal. Leaf size=197 \[ -\frac{5 e^2 \sqrt{d+e x}}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.113454, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {768, 646, 47, 63, 208} \[ -\frac{5 e^2 \sqrt{d+e x}}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 768
Rule 646
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{6 b}\\ &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac{\left (5 b e \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{\left (a b+b^2 x\right )^3} \, dx}{6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 e^2 \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{\left (a b+b^2 x\right )^2} \, dx}{8 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{16 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 e^2 \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(d+e x)^{5/2}}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{5 e^2 \sqrt{d+e x}}{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (d+e x)^{3/2}}{12 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^3 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{7/2} \sqrt{b d-a e} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.163273, size = 132, normalized size = 0.67 \[ \frac{\frac{15 e^3 (a+b x)^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{\sqrt{a e-b d}}-\sqrt{b} \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (d+4 e x)+b^2 \left (8 d^2+26 d e x+33 e^2 x^2\right )\right )}{24 b^{7/2} \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 316, normalized size = 1.6 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{2}}{24\,{b}^{3}} \left ( -15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{3}{b}^{3}{e}^{3}-45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){x}^{2}a{b}^{2}{e}^{3}+33\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{5/2}{b}^{2}-45\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) x{a}^{2}b{e}^{3}+40\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}abe-40\,\sqrt{ \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{3/2}{b}^{2}d-15\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ){a}^{3}{e}^{3}+15\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{a}^{2}{e}^{2}-30\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}abde+15\,\sqrt{ \left ( ae-bd \right ) b}\sqrt{ex+d}{b}^{2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \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{{\left (b x + a\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.0809, size = 1162, normalized size = 5.9 \begin{align*} \left [\frac{15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e +{\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\right )}}, \frac{15 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (8 \, b^{4} d^{3} + 2 \, a b^{3} d^{2} e + 5 \, a^{2} b^{2} d e^{2} - 15 \, a^{3} b e^{3} + 33 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} d^{2} e + 7 \, a b^{3} d e^{2} - 20 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (a^{3} b^{5} d - a^{4} b^{4} e +{\left (b^{8} d - a b^{7} e\right )} x^{3} + 3 \,{\left (a b^{7} d - a^{2} b^{6} e\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d - a^{3} b^{5} e\right )} x\right )}}\right ] \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 = 1.25684, size = 288, normalized size = 1.46 \begin{align*} \frac{5 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \, \sqrt{-b^{2} d + a b e} b^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac{33 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} - 40 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} + 15 \, \sqrt{x e + d} b^{2} d^{2} e^{3} + 40 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} - 30 \, \sqrt{x e + d} a b d e^{4} + 15 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{3} \mathrm{sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]