3.1572 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=149 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*(b*d - a*e)^2*(d + e*x)^7) + (b^2*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*(b*d - a*e)^3*(d + e*x)^6)

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Rubi [A]  time = 0.154624, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*(b*d - a*e)^2*(d + e*x)^7) + (b^2*(a
+ b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*(b*d - a*e)^3*(d + e*x)^6)

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Rubi in Sympy [A]  time = 19.8783, size = 131, normalized size = 0.88 \[ - \frac{b e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{168 \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 \left (d + e x\right )^{8} \left (a e - b d\right )} \]

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)**9,x)

[Out]

-b*e*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(168*(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)**7*(a*e - b*d)**2
) - (2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(16*(d + e*x)**8*(a*e - b*
d))

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Mathematica [A]  time = 0.154394, size = 223, normalized size = 1.5 \[ -\frac{\sqrt{(a+b x)^2} \left (21 a^5 e^5+15 a^4 b e^4 (d+8 e x)+10 a^3 b^2 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a^2 b^3 e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a b^4 e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+b^5 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{168 e^6 (a+b x) (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(21*a^5*e^5 + 15*a^4*b*e^4*(d + 8*e*x) + 10*a^3*b^2*e^3*(d^2
+ 8*d*e*x + 28*e^2*x^2) + 6*a^2*b^3*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^
3*x^3) + 3*a*b^4*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4
) + b^5*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e
^5*x^5)))/(168*e^6*(a + b*x)*(d + e*x)^8)

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Maple [B]  time = 0.015, size = 288, normalized size = 1.9 \[ -{\frac{56\,{x}^{5}{b}^{5}{e}^{5}+210\,{x}^{4}a{b}^{4}{e}^{5}+70\,{x}^{4}{b}^{5}d{e}^{4}+336\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+168\,{x}^{3}a{b}^{4}d{e}^{4}+56\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+280\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+168\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+84\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+28\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+120\,x{a}^{4}b{e}^{5}+80\,x{a}^{3}{b}^{2}d{e}^{4}+48\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+24\,xa{b}^{4}{d}^{3}{e}^{2}+8\,x{b}^{5}{d}^{4}e+21\,{a}^{5}{e}^{5}+15\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+6\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{168\,{e}^{6} \left ( ex+d \right ) ^{8} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^9,x)

[Out]

-1/168/e^6*(56*b^5*e^5*x^5+210*a*b^4*e^5*x^4+70*b^5*d*e^4*x^4+336*a^2*b^3*e^5*x^
3+168*a*b^4*d*e^4*x^3+56*b^5*d^2*e^3*x^3+280*a^3*b^2*e^5*x^2+168*a^2*b^3*d*e^4*x
^2+84*a*b^4*d^2*e^3*x^2+28*b^5*d^3*e^2*x^2+120*a^4*b*e^5*x+80*a^3*b^2*d*e^4*x+48
*a^2*b^3*d^2*e^3*x+24*a*b^4*d^3*e^2*x+8*b^5*d^4*e*x+21*a^5*e^5+15*a^4*b*d*e^4+10
*a^3*b^2*d^2*e^3+6*a^2*b^3*d^3*e^2+3*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x
+d)^8/(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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209806, size = 455, normalized size = 3.05 \[ -\frac{56 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 3 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} + 21 \, a^{5} e^{5} + 70 \,{\left (b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 56 \,{\left (b^{5} d^{2} e^{3} + 3 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} + 28 \,{\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 8 \,{\left (b^{5} d^{4} e + 3 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 15 \, a^{4} b e^{5}\right )} x}{168 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]

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)^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*e^5*x^5 + b^5*d^5 + 3*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^
2*d^2*e^3 + 15*a^4*b*d*e^4 + 21*a^5*e^5 + 70*(b^5*d*e^4 + 3*a*b^4*e^5)*x^4 + 56*
(b^5*d^2*e^3 + 3*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 + 28*(b^5*d^3*e^2 + 3*a*b^4*d^
2*e^3 + 6*a^2*b^3*d*e^4 + 10*a^3*b^2*e^5)*x^2 + 8*(b^5*d^4*e + 3*a*b^4*d^3*e^2 +
6*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + 15*a^4*b*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^
7 + 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^
6*e^8*x^2 + 8*d^7*e^7*x + d^8*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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221203, size = 514, normalized size = 3.45 \[ -\frac{{\left (56 \, b^{5} x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 70 \, b^{5} d x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 56 \, b^{5} d^{2} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 28 \, b^{5} d^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 8 \, b^{5} d^{4} x e{\rm sign}\left (b x + a\right ) + b^{5} d^{5}{\rm sign}\left (b x + a\right ) + 210 \, a b^{4} x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 168 \, a b^{4} d x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 84 \, a b^{4} d^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 24 \, a b^{4} d^{3} x e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b^{4} d^{4} e{\rm sign}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 168 \, a^{2} b^{3} d x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 48 \, a^{2} b^{3} d^{2} x e^{3}{\rm sign}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{3} e^{2}{\rm sign}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 80 \, a^{3} b^{2} d x e^{4}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 120 \, a^{4} b x e^{5}{\rm sign}\left (b x + a\right ) + 15 \, a^{4} b d e^{4}{\rm sign}\left (b x + a\right ) + 21 \, a^{5} e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-6\right )}}{168 \,{\left (x e + d\right )}^{8}} \]

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)^9,x, algorithm="giac")

[Out]

-1/168*(56*b^5*x^5*e^5*sign(b*x + a) + 70*b^5*d*x^4*e^4*sign(b*x + a) + 56*b^5*d
^2*x^3*e^3*sign(b*x + a) + 28*b^5*d^3*x^2*e^2*sign(b*x + a) + 8*b^5*d^4*x*e*sign
(b*x + a) + b^5*d^5*sign(b*x + a) + 210*a*b^4*x^4*e^5*sign(b*x + a) + 168*a*b^4*
d*x^3*e^4*sign(b*x + a) + 84*a*b^4*d^2*x^2*e^3*sign(b*x + a) + 24*a*b^4*d^3*x*e^
2*sign(b*x + a) + 3*a*b^4*d^4*e*sign(b*x + a) + 336*a^2*b^3*x^3*e^5*sign(b*x + a
) + 168*a^2*b^3*d*x^2*e^4*sign(b*x + a) + 48*a^2*b^3*d^2*x*e^3*sign(b*x + a) + 6
*a^2*b^3*d^3*e^2*sign(b*x + a) + 280*a^3*b^2*x^2*e^5*sign(b*x + a) + 80*a^3*b^2*
d*x*e^4*sign(b*x + a) + 10*a^3*b^2*d^2*e^3*sign(b*x + a) + 120*a^4*b*x*e^5*sign(
b*x + a) + 15*a^4*b*d*e^4*sign(b*x + a) + 21*a^5*e^5*sign(b*x + a))*e^(-6)/(x*e
+ d)^8