3.1573 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=200 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{84 (d+e x)^7 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{24 (d+e x)^8 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{9 (d+e x)^9 (b d-a e)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{504 (d+e x)^6 (b d-a e)^4} \]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(24*(b*d - a*e)^2*(d + e*x)^8) + (b^2*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(84*(b*d - a*e)^3*(d + e*x)^7) + (b^3*(
a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*(b*d - a*e)^4*(d + e*x)^6)

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Rubi [A]  time = 0.202687, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{84 (d+e x)^7 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{24 (d+e x)^8 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{9 (d+e x)^9 (b d-a e)}+\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{504 (d+e x)^6 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(24*(b*d - a*e)^2*(d + e*x)^8) + (b^2*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(84*(b*d - a*e)^3*(d + e*x)^7) + (b^3*(
a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*(b*d - a*e)^4*(d + e*x)^6)

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Rubi in Sympy [A]  time = 32.7457, size = 182, normalized size = 0.91 \[ \frac{b^{2} e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{504 \left (d + e x\right )^{7} \left (a e - b d\right )^{4}} - \frac{b^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{144 \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 \left (d + e x\right )^{8} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 \left (d + e x\right )^{9} \left (a e - b d\right )} \]

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)**10,x)

[Out]

b**2*e*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(504*(d + e*x)**7*(a*e - b*d)**4) - b
**2*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(144*(d + e*x)**7*(a*e - b
*d)**3) + b*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(48*(d + e*x)**8*(
a*e - b*d)**2) - (2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(18*(d + e*x)
**9*(a*e - b*d))

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Mathematica [A]  time = 0.171375, size = 223, normalized size = 1.12 \[ -\frac{\sqrt{(a+b x)^2} \left (56 a^5 e^5+35 a^4 b e^4 (d+9 e x)+20 a^3 b^2 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^2 b^3 e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 a b^4 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+b^5 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{504 e^6 (a+b x) (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(56*a^5*e^5 + 35*a^4*b*e^4*(d + 9*e*x) + 20*a^3*b^2*e^3*(d^2
+ 9*d*e*x + 36*e^2*x^2) + 10*a^2*b^3*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e
^3*x^3) + 4*a*b^4*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x
^4) + b^5*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 1
26*e^5*x^5)))/(504*e^6*(a + b*x)*(d + e*x)^9)

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Maple [A]  time = 0.015, size = 288, normalized size = 1.4 \[ -{\frac{126\,{x}^{5}{b}^{5}{e}^{5}+504\,{x}^{4}a{b}^{4}{e}^{5}+126\,{x}^{4}{b}^{5}d{e}^{4}+840\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+336\,{x}^{3}a{b}^{4}d{e}^{4}+84\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+720\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+360\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+144\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+36\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+315\,x{a}^{4}b{e}^{5}+180\,x{a}^{3}{b}^{2}d{e}^{4}+90\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+36\,xa{b}^{4}{d}^{3}{e}^{2}+9\,x{b}^{5}{d}^{4}e+56\,{a}^{5}{e}^{5}+35\,{a}^{4}bd{e}^{4}+20\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+4\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{504\,{e}^{6} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^10,x)

[Out]

-1/504/e^6*(126*b^5*e^5*x^5+504*a*b^4*e^5*x^4+126*b^5*d*e^4*x^4+840*a^2*b^3*e^5*
x^3+336*a*b^4*d*e^4*x^3+84*b^5*d^2*e^3*x^3+720*a^3*b^2*e^5*x^2+360*a^2*b^3*d*e^4
*x^2+144*a*b^4*d^2*e^3*x^2+36*b^5*d^3*e^2*x^2+315*a^4*b*e^5*x+180*a^3*b^2*d*e^4*
x+90*a^2*b^3*d^2*e^3*x+36*a*b^4*d^3*e^2*x+9*b^5*d^4*e*x+56*a^5*e^5+35*a^4*b*d*e^
4+20*a^3*b^2*d^2*e^3+10*a^2*b^3*d^3*e^2+4*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)
/(e*x+d)^9/(b*x+a)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209156, size = 470, normalized size = 2.35 \[ -\frac{126 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 4 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} + 20 \, a^{3} b^{2} d^{2} e^{3} + 35 \, a^{4} b d e^{4} + 56 \, a^{5} e^{5} + 126 \,{\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 84 \,{\left (b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} + 36 \,{\left (b^{5} d^{3} e^{2} + 4 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} + 20 \, a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x}{504 \,{\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*e^5*x^5 + b^5*d^5 + 4*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 + 20*a^3*
b^2*d^2*e^3 + 35*a^4*b*d*e^4 + 56*a^5*e^5 + 126*(b^5*d*e^4 + 4*a*b^4*e^5)*x^4 +
84*(b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 10*a^2*b^3*e^5)*x^3 + 36*(b^5*d^3*e^2 + 4*a*b^
4*d^2*e^3 + 10*a^2*b^3*d*e^4 + 20*a^3*b^2*e^5)*x^2 + 9*(b^5*d^4*e + 4*a*b^4*d^3*
e^2 + 10*a^2*b^3*d^2*e^3 + 20*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/(e^15*x^9 + 9*d*e
^14*x^8 + 36*d^2*e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^
4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*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)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223298, size = 514, normalized size = 2.57 \[ -\frac{{\left (126 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 126 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 84 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 36 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 9 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 504 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 336 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 144 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 36 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + 4 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 840 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 360 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 90 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 720 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 180 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 315 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + 35 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 56 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{504 \,{\left (x e + d\right )}^{9}} \]

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)^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5*e^5*sign(b*x + a) + 126*b^5*d*x^4*e^4*sign(b*x + a) + 84*b^5
*d^2*x^3*e^3*sign(b*x + a) + 36*b^5*d^3*x^2*e^2*sign(b*x + a) + 9*b^5*d^4*x*e*si
gn(b*x + a) + b^5*d^5*sign(b*x + a) + 504*a*b^4*x^4*e^5*sign(b*x + a) + 336*a*b^
4*d*x^3*e^4*sign(b*x + a) + 144*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 36*a*b^4*d^3*x
*e^2*sign(b*x + a) + 4*a*b^4*d^4*e*sign(b*x + a) + 840*a^2*b^3*x^3*e^5*sign(b*x
+ a) + 360*a^2*b^3*d*x^2*e^4*sign(b*x + a) + 90*a^2*b^3*d^2*x*e^3*sign(b*x + a)
+ 10*a^2*b^3*d^3*e^2*sign(b*x + a) + 720*a^3*b^2*x^2*e^5*sign(b*x + a) + 180*a^3
*b^2*d*x*e^4*sign(b*x + a) + 20*a^3*b^2*d^2*e^3*sign(b*x + a) + 315*a^4*b*x*e^5*
sign(b*x + a) + 35*a^4*b*d*e^4*sign(b*x + a) + 56*a^5*e^5*sign(b*x + a))*e^(-6)/
(x*e + d)^9