3.1701 \(\int \frac{(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{(d+e x)^{7/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) \sqrt{d+e x} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(35*e^2*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) + (35*e^2*(a + b*x)*(d + e*x)^(3/2))/(12*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(7*e*(d + e*x)^(5/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(7/2)/(
2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(b*d - a*e)^(3/2)*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(9/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])

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Rubi [A]  time = 0.386019, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(d+e x)^{7/2}}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (d+e x)^{5/2}}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 e^2 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) \sqrt{d+e x} (b d-a e)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^2 (a+b x) (d+e x)^{3/2}}{12 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(35*e^2*(b*d - a*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) + (35*e^2*(a + b*x)*(d + e*x)^(3/2))/(12*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(7*e*(d + e*x)^(5/2))/(4*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (d + e*x)^(7/2)/(
2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^2*(b*d - a*e)^(3/2)*(a + b*
x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(9/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.442728, size = 150, normalized size = 0.59 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (8 e^2 (10 b d-9 a e)-\frac{39 e (b d-a e)^2}{a+b x}-\frac{6 (b d-a e)^3}{(a+b x)^2}+8 b e^3 x\right )}{3 b^4}-\frac{35 e^2 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*(8*e^2*(10*b*d - 9*a*e) + 8*b*e^3*x - (6*(b*d - a*e
)^3)/(a + b*x)^2 - (39*e*(b*d - a*e)^2)/(a + b*x)))/(3*b^4) - (35*e^2*(b*d - a*e
)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)))/(4*((a + b*x
)^2)^(3/2))

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Maple [B]  time = 0.026, size = 714, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/12*(8*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x^2*b^3*e^2+105*arctan((e*x+d)^(1/2)*b
/(b*(a*e-b*d))^(1/2))*x^2*a^2*b^2*e^4-210*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(
1/2))*x^2*a*b^3*d*e^3+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x^2*b^4*d^
2*e^2+16*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*x*a*b^2*e^2-72*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*x^2*a*b^2*e^3+72*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x^2*b^3*d*e^2+210
*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*x*a^3*b*e^4-420*arctan((e*x+d)^(1/2
)*b/(b*(a*e-b*d))^(1/2))*x*a^2*b^2*d*e^3+210*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d)
)^(1/2))*x*a*b^3*d^2*e^2-31*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2+78*(b*(a
*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e-39*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*
d^2-144*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*x*a^2*b*e^3+144*(b*(a*e-b*d))^(1/2)*(e
*x+d)^(1/2)*x*a*b^2*d*e^2+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^4*e^
4-210*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^3*b*d*e^3+105*arctan((e*x+d)
^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^2*b^2*d^2*e^2-105*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1
/2)*a^3*e^3+171*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^2*b*d*e^2-99*(b*(a*e-b*d))^(
1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e+33*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)*(b*
x+a)/(b*(a*e-b*d))^(1/2)/b^4/((b*x+a)^2)^(3/2)

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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((e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222108, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (a^{2} b d e^{2} - a^{3} e^{3} +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \,{\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} -{\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, -\frac{105 \,{\left (a^{2} b d e^{2} - a^{3} e^{3} +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (8 \, b^{3} e^{3} x^{3} - 6 \, b^{3} d^{3} - 21 \, a b^{2} d^{2} e + 140 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 8 \,{\left (10 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} -{\left (39 \, b^{3} d^{2} e - 238 \, a b^{2} d e^{2} + 175 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/24*(105*(a^2*b*d*e^2 - a^3*e^3 + (b^3*d*e^2 - a*b^2*e^3)*x^2 + 2*(a*b^2*d*e^
2 - a^2*b*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)
*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(8*b^3*e^3*x^3 - 6*b^3*d^3 - 21*a*b^2*d^2
*e + 140*a^2*b*d*e^2 - 105*a^3*e^3 + 8*(10*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 - (39*b^
3*d^2*e - 238*a*b^2*d*e^2 + 175*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*
x + a^2*b^4), -1/12*(105*(a^2*b*d*e^2 - a^3*e^3 + (b^3*d*e^2 - a*b^2*e^3)*x^2 +
2*(a*b^2*d*e^2 - a^2*b*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(
b*d - a*e)/b)) - (8*b^3*e^3*x^3 - 6*b^3*d^3 - 21*a*b^2*d^2*e + 140*a^2*b*d*e^2 -
105*a^3*e^3 + 8*(10*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 - (39*b^3*d^2*e - 238*a*b^2*d*
e^2 + 175*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

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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((e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24253, size = 459, normalized size = 1.81 \[ -\frac{35 \,{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} + \frac{13 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{2} - 11 \, \sqrt{x e + d} b^{3} d^{3} e^{2} - 26 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{3} + 33 \, \sqrt{x e + d} a b^{2} d^{2} e^{3} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{4} - 33 \, \sqrt{x e + d} a^{2} b d e^{4} + 11 \, \sqrt{x e + d} a^{3} e^{5}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} - \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{6} e^{2} + 9 \, \sqrt{x e + d} b^{6} d e^{2} - 9 \, \sqrt{x e + d} a b^{5} e^{3}\right )}}{3 \, b^{9}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

-35/4*(b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d +
a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) + 1/4*(
13*(x*e + d)^(3/2)*b^3*d^2*e^2 - 11*sqrt(x*e + d)*b^3*d^3*e^2 - 26*(x*e + d)^(3/
2)*a*b^2*d*e^3 + 33*sqrt(x*e + d)*a*b^2*d^2*e^3 + 13*(x*e + d)^(3/2)*a^2*b*e^4 -
33*sqrt(x*e + d)*a^2*b*d*e^4 + 11*sqrt(x*e + d)*a^3*e^5)/(((x*e + d)*b - b*d +
a*e)^2*b^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2)) - 2/3*((x*e + d)^(3/2)*b^6*e^2
+ 9*sqrt(x*e + d)*b^6*d*e^2 - 9*sqrt(x*e + d)*a*b^5*e^3)/(b^9*sign(-(x*e + d)*b*
e + b*d*e - a*e^2))