3.1593 \(\int \frac{(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-20*e^3*(b*d - a*e)^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*d - a*e)^6/(4*b
^7*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*e*(b*d - a*e)^5)/(b^7*(a + b*
x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*e^2*(b*d - a*e)^4)/(2*b^7*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) + (e^5*(6*b*d - 5*a*e)*x*(a + b*x))/(b^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (e^6*x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (15*e^4*(b*d - a*e)^2*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])

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Rubi [A]  time = 0.58308, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{15 e^4 (a+b x) (b d-a e)^2 \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{20 e^3 (b d-a e)^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 e^2 (b d-a e)^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 e (b d-a e)^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x) (6 b d-5 a e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^6 x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-20*e^3*(b*d - a*e)^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*d - a*e)^6/(4*b
^7*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*e*(b*d - a*e)^5)/(b^7*(a + b*
x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*e^2*(b*d - a*e)^4)/(2*b^7*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) + (e^5*(6*b*d - 5*a*e)*x*(a + b*x))/(b^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (e^6*x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (15*e^4*(b*d - a*e)^2*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])

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Rubi in Sympy [A]  time = 53.929, size = 284, normalized size = 0.94 \[ - \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{6}}{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{e \left (d + e x\right )^{5}}{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 e^{2} \left (2 a + 2 b x\right ) \left (d + e x\right )^{4}}{8 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{5 e^{3} \left (d + e x\right )^{3}}{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{15 e^{4} \left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{15 e^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{7}} + \frac{15 e^{4} \left (a + b x\right ) \left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{7} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

-(2*a + 2*b*x)*(d + e*x)**6/(8*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) - e*(d + e
*x)**5/(2*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - 5*e**2*(2*a + 2*b*x)*(d +
e*x)**4/(8*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) - 5*e**3*(d + e*x)**3/(b**4
*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + 15*e**4*(2*a + 2*b*x)*(d + e*x)**2/(4*b**5*
sqrt(a**2 + 2*a*b*x + b**2*x**2)) - 15*e**5*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b*
*2*x**2)/b**7 + 15*e**4*(a + b*x)*(a*e - b*d)**2*log(a + b*x)/(b**7*sqrt(a**2 +
2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.294074, size = 313, normalized size = 1.04 \[ \frac{57 a^6 e^6+14 a^5 b e^5 (12 e x-11 d)+a^4 b^2 e^4 \left (125 d^2-496 d e x+132 e^2 x^2\right )-4 a^3 b^3 e^3 \left (5 d^3-110 d^2 e x+126 d e^2 x^2+8 e^3 x^3\right )-a^2 b^4 e^2 \left (5 d^4+80 d^3 e x-540 d^2 e^2 x^2+96 d e^3 x^3+68 e^4 x^4\right )-2 a b^5 e \left (d^5+10 d^4 e x+60 d^3 e^2 x^2-120 d^2 e^3 x^3-48 d e^4 x^4+6 e^5 x^5\right )+60 e^4 (a+b x)^4 (b d-a e)^2 \log (a+b x)+b^6 \left (-\left (d^6+8 d^5 e x+30 d^4 e^2 x^2+80 d^3 e^3 x^3-24 d e^5 x^5-2 e^6 x^6\right )\right )}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(57*a^6*e^6 + 14*a^5*b*e^5*(-11*d + 12*e*x) + a^4*b^2*e^4*(125*d^2 - 496*d*e*x +
132*e^2*x^2) - 4*a^3*b^3*e^3*(5*d^3 - 110*d^2*e*x + 126*d*e^2*x^2 + 8*e^3*x^3)
- a^2*b^4*e^2*(5*d^4 + 80*d^3*e*x - 540*d^2*e^2*x^2 + 96*d*e^3*x^3 + 68*e^4*x^4)
- 2*a*b^5*e*(d^5 + 10*d^4*e*x + 60*d^3*e^2*x^2 - 120*d^2*e^3*x^3 - 48*d*e^4*x^4
+ 6*e^5*x^5) - b^6*(d^6 + 8*d^5*e*x + 30*d^4*e^2*x^2 + 80*d^3*e^3*x^3 - 24*d*e^
5*x^5 - 2*e^6*x^6) + 60*e^4*(b*d - a*e)^2*(a + b*x)^4*Log[a + b*x])/(4*b^7*(a +
b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.027, size = 661, normalized size = 2.2 \[{\frac{ \left ( -480\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}-720\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-480\,\ln \left ( bx+a \right ) x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( bx+a \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+57\,{a}^{6}{e}^{6}-{b}^{6}{d}^{6}-12\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-80\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-32\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-68\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+132\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+168\,x{a}^{5}b{e}^{6}-8\,x{b}^{6}{d}^{5}e-496\,x{a}^{4}{b}^{2}d{e}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}{e}^{6}+240\,\ln \left ( bx+a \right ) x{a}^{5}b{e}^{6}-120\,\ln \left ( bx+a \right ){a}^{5}bd{e}^{5}+60\,\ln \left ( bx+a \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}+540\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-120\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-2\,{d}^{5}a{b}^{5}e-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-5\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+60\,\ln \left ( bx+a \right ){a}^{6}{e}^{6}-120\,\ln \left ( bx+a \right ){x}^{4}a{b}^{5}d{e}^{5}+440\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-80\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-20\,xa{b}^{5}{d}^{4}{e}^{2}+240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}{e}^{6}+60\,\ln \left ( bx+a \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}-504\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+96\,{x}^{4}a{b}^{5}d{e}^{5}-96\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+125\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-154\,{a}^{5}bd{e}^{5}+2\,{x}^{6}{b}^{6}{e}^{6} \right ) \left ( bx+a \right ) }{4\,{b}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/4*(-480*ln(b*x+a)*x^3*a^2*b^4*d*e^5+240*ln(b*x+a)*x^3*a*b^5*d^2*e^4-720*ln(b*x
+a)*x^2*a^3*b^3*d*e^5+360*ln(b*x+a)*x^2*a^2*b^4*d^2*e^4-480*ln(b*x+a)*x*a^4*b^2*
d*e^5+240*ln(b*x+a)*x*a^3*b^3*d^2*e^4+57*a^6*e^6-b^6*d^6-12*x^5*a*b^5*e^6+24*x^5
*b^6*d*e^5-80*x^3*b^6*d^3*e^3-32*x^3*a^3*b^3*e^6-68*x^4*a^2*b^4*e^6+132*x^2*a^4*
b^2*e^6-30*x^2*b^6*d^4*e^2+168*x*a^5*b*e^6-8*x*b^6*d^5*e-496*x*a^4*b^2*d*e^5+240
*ln(b*x+a)*x^3*a^3*b^3*e^6+360*ln(b*x+a)*x^2*a^4*b^2*e^6+240*ln(b*x+a)*x*a^5*b*e
^6-120*ln(b*x+a)*a^5*b*d*e^5+60*ln(b*x+a)*a^4*b^2*d^2*e^4+540*x^2*a^2*b^4*d^2*e^
4-120*x^2*a*b^5*d^3*e^3-2*d^5*a*b^5*e-20*a^3*b^3*d^3*e^3-5*d^4*e^2*a^2*b^4+60*ln
(b*x+a)*a^6*e^6-120*ln(b*x+a)*x^4*a*b^5*d*e^5+440*x*a^3*b^3*d^2*e^4-80*x*a^2*b^4
*d^3*e^3-20*x*a*b^5*d^4*e^2+240*x^3*a*b^5*d^2*e^4+60*ln(b*x+a)*x^4*a^2*b^4*e^6+6
0*ln(b*x+a)*x^4*b^6*d^2*e^4-504*x^2*a^3*b^3*d*e^5+96*x^4*a*b^5*d*e^5-96*x^3*a^2*
b^4*d*e^5+125*b^2*a^4*d^2*e^4-154*a^5*b*d*e^5+2*x^6*b^6*e^6)*(b*x+a)/b^7/((b*x+a
)^2)^(5/2)

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Maxima [A]  time = 0.789447, size = 844, normalized size = 2.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

1/4*e^6*((2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b
^2*x^2 + 168*a^5*b*x + 57*a^6)/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*
b^8*x + a^4*b^7) + 60*a^2*log(b*x + a)/b^7) + 1/2*d*e^5*((12*b^5*x^5 + 48*a*b^4*
x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*b
^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 60*a*log(b*x + a)/b^6) + 5/4*d
^2*e^4*((48*a*b^3*x^3 + 108*a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^
8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*x + a)/b^5) - 5/3*d^3*
e^3*(12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x +
a^2)^(3/2)*b^4) + 3*a^3*b/((b^2)^(9/2)*(x + a/b)^4) - 8*a^2/((b^2)^(7/2)*(x + a/
b)^3) + 6*a/((b^2)^(5/2)*b*(x + a/b)^2) - 6*a^3/((b^2)^(5/2)*b^3*(x + a/b)^4)) -
1/2*d^5*e*(4/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/((b^2)^(5/2)*b*(x + a/
b)^4)) - 5/4*d^4*e^2*(3*a^2*b^2/((b^2)^(9/2)*(x + a/b)^4) - 8*a*b/((b^2)^(7/2)*(
x + a/b)^3) + 6/((b^2)^(5/2)*(x + a/b)^2)) - 1/4*d^6/((b^2)^(5/2)*(x + a/b)^4)

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Fricas [A]  time = 0.212642, size = 772, normalized size = 2.56 \[ \frac{2 \, b^{6} e^{6} x^{6} - b^{6} d^{6} - 2 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 125 \, a^{4} b^{2} d^{2} e^{4} - 154 \, a^{5} b d e^{5} + 57 \, a^{6} e^{6} + 12 \,{\left (2 \, b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 4 \,{\left (24 \, a b^{5} d e^{5} - 17 \, a^{2} b^{4} e^{6}\right )} x^{4} - 16 \,{\left (5 \, b^{6} d^{3} e^{3} - 15 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{3} - 6 \,{\left (5 \, b^{6} d^{4} e^{2} + 20 \, a b^{5} d^{3} e^{3} - 90 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} - 22 \, a^{4} b^{2} e^{6}\right )} x^{2} - 4 \,{\left (2 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 20 \, a^{2} b^{4} d^{3} e^{3} - 110 \, a^{3} b^{3} d^{2} e^{4} + 124 \, a^{4} b^{2} d e^{5} - 42 \, a^{5} b e^{6}\right )} x + 60 \,{\left (a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} + a^{6} e^{6} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (a b^{5} d^{2} e^{4} - 2 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (a^{2} b^{4} d^{2} e^{4} - 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (a^{3} b^{3} d^{2} e^{4} - 2 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 - b^6*d^6 - 2*a*b^5*d^5*e - 5*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^
3*e^3 + 125*a^4*b^2*d^2*e^4 - 154*a^5*b*d*e^5 + 57*a^6*e^6 + 12*(2*b^6*d*e^5 - a
*b^5*e^6)*x^5 + 4*(24*a*b^5*d*e^5 - 17*a^2*b^4*e^6)*x^4 - 16*(5*b^6*d^3*e^3 - 15
*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^3 - 6*(5*b^6*d^4*e^2 + 20*a*
b^5*d^3*e^3 - 90*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 - 22*a^4*b^2*e^6)*x^2 - 4*(2
*b^6*d^5*e + 5*a*b^5*d^4*e^2 + 20*a^2*b^4*d^3*e^3 - 110*a^3*b^3*d^2*e^4 + 124*a^
4*b^2*d*e^5 - 42*a^5*b*e^6)*x + 60*(a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 + a^6*e^6 +
(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 - 2*a^2*b^4*d
*e^5 + a^3*b^3*e^6)*x^3 + 6*(a^2*b^4*d^2*e^4 - 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^
2 + 4*(a^3*b^3*d^2*e^4 - 2*a^4*b^2*d*e^5 + a^5*b*e^6)*x)*log(b*x + a))/(b^11*x^4
+ 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{6}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.585388, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^6/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x