3.1645 \(\int \frac{(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=145 \[ -\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{35 e^3 \sqrt{d+e x}}{8 b^4} \]

[Out]

(35*e^3*Sqrt[d + e*x])/(8*b^4) - (35*e^2*(d + e*x)^(3/2))/(24*b^3*(a + b*x)) - (
7*e*(d + e*x)^(5/2))/(12*b^2*(a + b*x)^2) - (d + e*x)^(7/2)/(3*b*(a + b*x)^3) -
(35*e^3*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(
9/2))

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Rubi [A]  time = 0.208393, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}-\frac{35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac{7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac{(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac{35 e^3 \sqrt{d+e x}}{8 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(35*e^3*Sqrt[d + e*x])/(8*b^4) - (35*e^2*(d + e*x)^(3/2))/(24*b^3*(a + b*x)) - (
7*e*(d + e*x)^(5/2))/(12*b^2*(a + b*x)^2) - (d + e*x)^(7/2)/(3*b*(a + b*x)^3) -
(35*e^3*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(
9/2))

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Rubi in Sympy [A]  time = 53.6202, size = 131, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{3 b \left (a + b x\right )^{3}} - \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{12 b^{2} \left (a + b x\right )^{2}} - \frac{35 e^{2} \left (d + e x\right )^{\frac{3}{2}}}{24 b^{3} \left (a + b x\right )} + \frac{35 e^{3} \sqrt{d + e x}}{8 b^{4}} - \frac{35 e^{3} \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 b^{\frac{9}{2}}} \]

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)**2,x)

[Out]

-(d + e*x)**(7/2)/(3*b*(a + b*x)**3) - 7*e*(d + e*x)**(5/2)/(12*b**2*(a + b*x)**
2) - 35*e**2*(d + e*x)**(3/2)/(24*b**3*(a + b*x)) + 35*e**3*sqrt(d + e*x)/(8*b**
4) - 35*e**3*sqrt(a*e - b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*b**(
9/2))

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Mathematica [A]  time = 0.246184, size = 139, normalized size = 0.96 \[ -\frac{35 e^3 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{9/2}}-\frac{\sqrt{d+e x} \left (87 e^2 (a+b x)^2 (b d-a e)+38 e (a+b x) (b d-a e)^2+8 (b d-a e)^3-48 e^3 (a+b x)^3\right )}{24 b^4 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(8*(b*d - a*e)^3 + 38*e*(b*d - a*e)^2*(a + b*x) + 87*e^2*(b*d -
a*e)*(a + b*x)^2 - 48*e^3*(a + b*x)^3))/(24*b^4*(a + b*x)^3) - (35*e^3*Sqrt[b*d
- a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2))

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Maple [B]  time = 0.028, size = 352, normalized size = 2.4 \[ 2\,{\frac{{e}^{3}\sqrt{ex+d}}{{b}^{4}}}+{\frac{29\,{e}^{4}a}{8\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{29\,{e}^{3}d}{8\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{17\,{a}^{2}{e}^{5}}{3\,{b}^{3} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{34\,{e}^{4}ad}{3\,{b}^{2} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{e}^{3}{d}^{2}}{3\,b \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{19\,{a}^{3}{e}^{6}}{8\,{b}^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{57\,{a}^{2}{e}^{5}d}{8\,{b}^{3} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{57\,{e}^{4}a{d}^{2}}{8\,{b}^{2} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{19\,{e}^{3}{d}^{3}}{8\,b \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{35\,{e}^{4}a}{8\,{b}^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{35\,{e}^{3}d}{8\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

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)^2,x)

[Out]

2*e^3*(e*x+d)^(1/2)/b^4+29/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*a-29/8*e^3/b/(b
*e*x+a*e)^3*(e*x+d)^(5/2)*d+17/3*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(3/2)*a^2-34/3*e^
4/b^2/(b*e*x+a*e)^3*(e*x+d)^(3/2)*a*d+17/3*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(3/2)*d^2
+19/8*e^6/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^3-57/8*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)
^(1/2)*a^2*d+57/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a*d^2-19/8*e^3/b/(b*e*x+a*
e)^3*(e*x+d)^(1/2)*d^3-35/8*e^4/b^4/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(
b*(a*e-b*d))^(1/2))*a+35/8*e^3/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b
*(a*e-b*d))^(1/2))*d

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222412, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac{105 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \,{\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[1/48*(105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*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*(48*b^3*e^3*x^3 - 8*b^3*d^3 - 14*a*b^2*d^2*e - 35*a^2*b*d*e^2 + 105*a^
3*e^3 - 3*(29*b^3*d*e^2 - 77*a*b^2*e^3)*x^2 - 2*(19*b^3*d^2*e + 49*a*b^2*d*e^2 -
140*a^2*b*e^3)*x)*sqrt(e*x + d))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4
), -1/24*(105*(b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*sqrt(-(b
*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (48*b^3*e^3*x^3 - 8*b^
3*d^3 - 14*a*b^2*d^2*e - 35*a^2*b*d*e^2 + 105*a^3*e^3 - 3*(29*b^3*d*e^2 - 77*a*b
^2*e^3)*x^2 - 2*(19*b^3*d^2*e + 49*a*b^2*d*e^2 - 140*a^2*b*e^3)*x)*sqrt(e*x + d)
)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224401, size = 335, normalized size = 2.31 \[ \frac{35 \,{\left (b d e^{3} - a e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{8 \, \sqrt{-b^{2} d + a b e} b^{4}} + \frac{2 \, \sqrt{x e + d} e^{3}}{b^{4}} - \frac{87 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} d e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d^{2} e^{3} + 57 \, \sqrt{x e + d} b^{3} d^{3} e^{3} - 87 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{2} e^{4} + 272 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} d e^{4} - 171 \, \sqrt{x e + d} a b^{2} d^{2} e^{4} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b e^{5} + 171 \, \sqrt{x e + d} a^{2} b d e^{5} - 57 \, \sqrt{x e + d} a^{3} e^{6}}{24 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}} \]

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)^2,x, algorithm="giac")

[Out]

35/8*(b*d*e^3 - a*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d
+ a*b*e)*b^4) + 2*sqrt(x*e + d)*e^3/b^4 - 1/24*(87*(x*e + d)^(5/2)*b^3*d*e^3 -
136*(x*e + d)^(3/2)*b^3*d^2*e^3 + 57*sqrt(x*e + d)*b^3*d^3*e^3 - 87*(x*e + d)^(5
/2)*a*b^2*e^4 + 272*(x*e + d)^(3/2)*a*b^2*d*e^4 - 171*sqrt(x*e + d)*a*b^2*d^2*e^
4 - 136*(x*e + d)^(3/2)*a^2*b*e^5 + 171*sqrt(x*e + d)*a^2*b*d*e^5 - 57*sqrt(x*e
+ d)*a^3*e^6)/(((x*e + d)*b - b*d + a*e)^3*b^4)