Optimal. Leaf size=318 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0943637, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 \sqrt{d+e x} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5 \sqrt{d+e x}}{e^5}+\frac{5 b^6 (b d-a e)^4 (d+e x)^{3/2}}{e^5}-\frac{10 b^7 (b d-a e)^3 (d+e x)^{5/2}}{e^5}+\frac{10 b^8 (b d-a e)^2 (d+e x)^{7/2}}{e^5}-\frac{5 b^9 (b d-a e) (d+e x)^{9/2}}{e^5}+\frac{b^{10} (d+e x)^{11/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^5 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac{2 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{20 b^2 (b d-a e)^3 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac{20 b^3 (b d-a e)^2 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac{10 b^4 (b d-a e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac{2 b^5 (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.123743, size = 235, normalized size = 0.74 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (286 a^2 b^3 e^2 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+858 a^3 b^2 e^3 \left (8 d^2-12 d e x+15 e^2 x^2\right )+3003 a^4 b e^4 (3 e x-2 d)+3003 a^5 e^5+13 a b^4 e \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-480 d^3 e^2 x^2+560 d^2 e^3 x^3+384 d^4 e x-256 d^5-630 d e^4 x^4+693 e^5 x^5\right )\right )}{9009 e^6 (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.155, size = 289, normalized size = 0.9 \begin{align*}{\frac{1386\,{x}^{5}{b}^{5}{e}^{5}+8190\,{x}^{4}a{b}^{4}{e}^{5}-1260\,{x}^{4}{b}^{5}d{e}^{4}+20020\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-7280\,{x}^{3}a{b}^{4}d{e}^{4}+1120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+25740\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-17160\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+6240\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+18018\,x{a}^{4}b{e}^{5}-20592\,x{a}^{3}{b}^{2}d{e}^{4}+13728\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-4992\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+6006\,{a}^{5}{e}^{5}-12012\,d{e}^{4}{a}^{4}b+13728\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-9152\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3328\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{9009\,{e}^{6} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \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 [A] time = 1.12311, size = 456, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \,{\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} +{\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6331, size = 761, normalized size = 2.39 \begin{align*} \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \,{\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} +{\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19161, size = 454, normalized size = 1.43 \begin{align*} \frac{2}{9009} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{4} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{3} b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a^{2} b^{3} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} a b^{4} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} b^{5} e^{\left (-5\right )} \mathrm{sgn}\left (b x + a\right ) + 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]