3.1683 \(\int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=318 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)} \]

[Out]

(-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
)) + (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)) - (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)) + (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)) - (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)) + (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))

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Rubi [A]  time = 0.294007, 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 \[ -\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^6 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}{3 e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-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
)) + (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)) - (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)) + (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)) - (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)) + (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))

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Rubi in Sympy [A]  time = 44.744, size = 274, normalized size = 0.86 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{13 e} + \frac{4 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{143 e^{2}} + \frac{160 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1287 e^{3}} + \frac{320 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{4}} + \frac{256 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{5}} + \frac{512 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{6} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

2*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(13*e) + 4*(5*a + 5*b*x)*
(d + e*x)**(3/2)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(143*e**2) + 16
0*(d + e*x)**(3/2)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(1287*e**3
) + 320*(3*a + 3*b*x)*(d + e*x)**(3/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2
*x**2)/(9009*e**4) + 256*(d + e*x)**(3/2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b
**2*x**2)/(3003*e**5) + 512*(d + e*x)**(3/2)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)/(9009*e**6*(a + b*x))

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Mathematica [A]  time = 0.286616, size = 235, normalized size = 0.74 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (3003 a^5 e^5+3003 a^4 b e^4 (3 e x-2 d)+858 a^3 b^2 e^3 \left (8 d^2-12 d e x+15 e^2 x^2\right )+286 a^2 b^3 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+13 a b^4 e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+b^5 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )\right )}{9009 e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(3003*a^5*e^5 + 3003*a^4*b*e^4*(-2*d + 3*e*
x) + 858*a^3*b^2*e^3*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 286*a^2*b^3*e^2*(-16*d^3
+ 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 13*a*b^4*e*(128*d^4 - 192*d^3*e*x +
240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + b^5*(-256*d^5 + 384*d^4*e*x - 4
80*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5)))/(9009*e^6*(a +
b*x))

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Maple [A]  time = 0.011, size = 289, normalized size = 0.9 \[{\frac{1386\,{x}^{5}{b}^{5}{e}^{5}+8190\,{x}^{4}a{b}^{4}{e}^{5}-1260\,{x}^{4}{b}^{5}d{e}^{4}+20020\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-7280\,{x}^{3}a{b}^{4}d{e}^{4}+1120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+25740\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-17160\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+6240\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-960\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+18018\,x{a}^{4}b{e}^{5}-20592\,x{a}^{3}{b}^{2}d{e}^{4}+13728\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-4992\,xa{b}^{4}{d}^{3}{e}^{2}+768\,x{b}^{5}{d}^{4}e+6006\,{a}^{5}{e}^{5}-12012\,{a}^{4}bd{e}^{4}+13728\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-9152\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3328\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{9009\,{e}^{6} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/9009*(e*x+d)^(3/2)*(693*b^5*e^5*x^5+4095*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+10010
*a^2*b^3*e^5*x^3-3640*a*b^4*d*e^4*x^3+560*b^5*d^2*e^3*x^3+12870*a^3*b^2*e^5*x^2-
8580*a^2*b^3*d*e^4*x^2+3120*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^2*x^2+9009*a^4*b*e^5
*x-10296*a^3*b^2*d*e^4*x+6864*a^2*b^3*d^2*e^3*x-2496*a*b^4*d^3*e^2*x+384*b^5*d^4
*e*x+3003*a^5*e^5-6006*a^4*b*d*e^4+6864*a^3*b^2*d^2*e^3-4576*a^2*b^3*d^3*e^2+166
4*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [A]  time = 0.738721, size = 456, normalized size = 1.43 \[ \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \,{\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} +{\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/9009*(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*(b^5*d*e^5 + 6
5*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^2*b^3*e^6)*x^4 + 1
0*(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)*x^3
- 3*(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)*x^2 + (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)*x)*sqrt(e*x + d)/e
^6

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Fricas [A]  time = 0.207686, size = 456, normalized size = 1.43 \[ \frac{2 \,{\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \,{\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} +{\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/9009*(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*(b^5*d*e^5 + 6
5*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^2*b^3*e^6)*x^4 + 1
0*(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)*x^3
- 3*(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)*x^2 + (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)*x)*sqrt(e*x + d)/e
^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)*((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.225802, size = 502, normalized size = 1.58 \[ \frac{2}{9009} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{4} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a^{3} b^{2} e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a^{2} b^{3} e^{\left (-27\right )}{\rm sign}\left (b x + a\right ) + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} a b^{4} e^{\left (-44\right )}{\rm sign}\left (b x + a\right ) +{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{60} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{60} + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{60} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{60} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{60} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{60}\right )} b^{5} e^{\left (-65\right )}{\rm sign}\left (b x + a\right ) + 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*sqrt(e*x + d),x, algorithm="giac")

[Out]

2/9009*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*b*e^(-1)*sign(b*x + a
) + 858*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2
)*d^2*e^12)*a^3*b^2*e^(-14)*sign(b*x + a) + 286*(35*(x*e + d)^(9/2)*e^24 - 135*(
x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e
^24)*a^2*b^3*e^(-27)*sign(b*x + a) + 13*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e +
d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40
+ 1155*(x*e + d)^(3/2)*d^4*e^40)*a*b^4*e^(-44)*sign(b*x + a) + (693*(x*e + d)^(
13/2)*e^60 - 4095*(x*e + d)^(11/2)*d*e^60 + 10010*(x*e + d)^(9/2)*d^2*e^60 - 128
70*(x*e + d)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(5/2)*d^4*e^60 - 3003*(x*e + d)^(3/
2)*d^5*e^60)*b^5*e^(-65)*sign(b*x + a) + 3003*(x*e + d)^(3/2)*a^5*sign(b*x + a))
*e^(-1)