3.1684 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=316 \[ -\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) +
(10*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a +
b*x)) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6
*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(7*e^6*(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*e^6*(a + b*x)) + (2*b^5*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^6*(a + b*x))

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Rubi [A]  time = 0.297411, antiderivative size = 316, 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 \[ -\frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{3 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^6 (a+b x)}+\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) +
(10*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a +
b*x)) - (4*b^2*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6
*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(7*e^6*(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(9*e^6*(a + b*x)) + (2*b^5*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^6*(a + b*x))

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Rubi in Sympy [A]  time = 44.7348, size = 274, normalized size = 0.87 \[ \frac{2 \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11 e} + \frac{4 \left (5 a + 5 b x\right ) \sqrt{d + e x} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 e^{2}} + \frac{160 \sqrt{d + e x} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{693 e^{3}} + \frac{64 \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{4}} + \frac{256 \sqrt{d + e x} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{5}} + \frac{512 \sqrt{d + e x} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{6} \left (a + b x\right )} \]

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*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(11*e) + 4*(5*a + 5*b*x)*sqr
t(d + e*x)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(99*e**2) + 160*sqrt(
d + e*x)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(693*e**3) + 64*(3*a
+ 3*b*x)*sqrt(d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*e**
4) + 256*sqrt(d + e*x)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*e**5
) + 512*sqrt(d + e*x)*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*e**6*
(a + b*x))

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Mathematica [A]  time = 0.202683, size = 234, normalized size = 0.74 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (693 a^5 e^5+1155 a^4 b e^4 (e x-2 d)+462 a^3 b^2 e^3 \left (8 d^2-4 d e x+3 e^2 x^2\right )+198 a^2 b^3 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+11 a b^4 e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )\right )}{693 e^6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(693*a^5*e^5 + 1155*a^4*b*e^4*(-2*d + e*x) +
462*a^3*b^2*e^3*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 198*a^2*b^3*e^2*(-16*d^3 + 8*d^2
*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 11*a*b^4*e*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*
x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + b^5*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2
+ 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5)))/(693*e^6*(a + b*x))

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Maple [A]  time = 0.01, size = 289, normalized size = 0.9 \[{\frac{126\,{x}^{5}{b}^{5}{e}^{5}+770\,{x}^{4}a{b}^{4}{e}^{5}-140\,{x}^{4}{b}^{5}d{e}^{4}+1980\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-880\,{x}^{3}a{b}^{4}d{e}^{4}+160\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+2772\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-2376\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+1056\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-192\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+2310\,x{a}^{4}b{e}^{5}-3696\,x{a}^{3}{b}^{2}d{e}^{4}+3168\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-1408\,xa{b}^{4}{d}^{3}{e}^{2}+256\,x{b}^{5}{d}^{4}e+1386\,{a}^{5}{e}^{5}-4620\,{a}^{4}bd{e}^{4}+7392\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-6336\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+2816\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{693\, \left ( bx+a \right ) ^{5}{e}^{6}}\sqrt{ex+d} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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]

2/693*(e*x+d)^(1/2)*(63*b^5*e^5*x^5+385*a*b^4*e^5*x^4-70*b^5*d*e^4*x^4+990*a^2*b
^3*e^5*x^3-440*a*b^4*d*e^4*x^3+80*b^5*d^2*e^3*x^3+1386*a^3*b^2*e^5*x^2-1188*a^2*
b^3*d*e^4*x^2+528*a*b^4*d^2*e^3*x^2-96*b^5*d^3*e^2*x^2+1155*a^4*b*e^5*x-1848*a^3
*b^2*d*e^4*x+1584*a^2*b^3*d^2*e^3*x-704*a*b^4*d^3*e^2*x+128*b^5*d^4*e*x+693*a^5*
e^5-2310*a^4*b*d*e^4+3696*a^3*b^2*d^2*e^3-3168*a^2*b^3*d^3*e^2+1408*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 = 0.740537, size = 456, normalized size = 1.44 \[ \frac{2 \,{\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \,{\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \,{\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} +{\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} -{\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )}}{693 \, \sqrt{e x + d} e^{6}} \]

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]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 +
3696*a^3*b^2*d^3*e^3 - 2310*a^4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*
b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b^3*e^6)*x^4 - 2*(8*b
^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^
5*d^4*e^2 - 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a
^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848
*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)/(sqrt(e*x + d)*e^6)

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Fricas [A]  time = 0.207846, size = 352, normalized size = 1.11 \[ \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 1408 \, a b^{4} d^{4} e - 3168 \, a^{2} b^{3} d^{3} e^{2} + 3696 \, a^{3} b^{2} d^{2} e^{3} - 2310 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 44 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 88 \, a b^{4} d^{2} e^{3} + 198 \, a^{2} b^{3} d e^{4} - 231 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 704 \, a b^{4} d^{3} e^{2} + 1584 \, a^{2} b^{3} d^{2} e^{3} - 1848 \, a^{3} b^{2} d e^{4} + 1155 \, a^{4} b e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{6}} \]

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]

2/693*(63*b^5*e^5*x^5 - 256*b^5*d^5 + 1408*a*b^4*d^4*e - 3168*a^2*b^3*d^3*e^2 +
3696*a^3*b^2*d^2*e^3 - 2310*a^4*b*d*e^4 + 693*a^5*e^5 - 35*(2*b^5*d*e^4 - 11*a*b
^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 44*a*b^4*d*e^4 + 99*a^2*b^3*e^5)*x^3 - 6*(16*b
^5*d^3*e^2 - 88*a*b^4*d^2*e^3 + 198*a^2*b^3*d*e^4 - 231*a^3*b^2*e^5)*x^2 + (128*
b^5*d^4*e - 704*a*b^4*d^3*e^2 + 1584*a^2*b^3*d^2*e^3 - 1848*a^3*b^2*d*e^4 + 1155
*a^4*b*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.219168, size = 500, normalized size = 1.58 \[ \frac{2}{693} \,{\left (1155 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{4} b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} a^{3} b^{2} e^{\left (-10\right )}{\rm sign}\left (b x + a\right ) + 198 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} a^{2} b^{3} e^{\left (-21\right )}{\rm sign}\left (b x + a\right ) + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} a b^{4} e^{\left (-36\right )}{\rm sign}\left (b x + a\right ) +{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} b^{5} e^{\left (-55\right )}{\rm sign}\left (b x + a\right ) + 693 \, \sqrt{x e + d} a^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

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]

2/693*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*e^(-1)*sign(b*x + a) + 4
62*(3*(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)
*a^3*b^2*e^(-10)*sign(b*x + a) + 198*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2
)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*a^2*b^3*e^(-
21)*sign(b*x + a) + 11*(35*(x*e + d)^(9/2)*e^32 - 180*(x*e + d)^(7/2)*d*e^32 + 3
78*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^32 + 315*sqrt(x*e + d)*d
^4*e^32)*a*b^4*e^(-36)*sign(b*x + a) + (63*(x*e + d)^(11/2)*e^50 - 385*(x*e + d)
^(9/2)*d*e^50 + 990*(x*e + d)^(7/2)*d^2*e^50 - 1386*(x*e + d)^(5/2)*d^3*e^50 + 1
155*(x*e + d)^(3/2)*d^4*e^50 - 693*sqrt(x*e + d)*d^5*e^50)*b^5*e^(-55)*sign(b*x
+ a) + 693*sqrt(x*e + d)*a^5*sign(b*x + a))*e^(-1)