3.1650 \(\int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{35 e^3}{8 \sqrt{d+e x} (b d-a e)^4}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}-\frac{35 e^2}{24 (a+b x) \sqrt{d+e x} (b d-a e)^3}+\frac{7 e}{12 (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

[Out]

(-35*e^3)/(8*(b*d - a*e)^4*Sqrt[d + e*x]) - 1/(3*(b*d - a*e)*(a + b*x)^3*Sqrt[d
+ e*x]) + (7*e)/(12*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) - (35*e^2)/(24*(b*d
- a*e)^3*(a + b*x)*Sqrt[d + e*x]) + (35*Sqrt[b]*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.270013, antiderivative size = 173, 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 \[ -\frac{35 e^3}{8 \sqrt{d+e x} (b d-a e)^4}+\frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}-\frac{35 e^2}{24 (a+b x) \sqrt{d+e x} (b d-a e)^3}+\frac{7 e}{12 (a+b x)^2 \sqrt{d+e x} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-35*e^3)/(8*(b*d - a*e)^4*Sqrt[d + e*x]) - 1/(3*(b*d - a*e)*(a + b*x)^3*Sqrt[d
+ e*x]) + (7*e)/(12*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]) - (35*e^2)/(24*(b*d
- a*e)^3*(a + b*x)*Sqrt[d + e*x]) + (35*Sqrt[b]*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2))

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Rubi in Sympy [A]  time = 80.3693, size = 162, normalized size = 0.94 \[ - \frac{35 \sqrt{b} e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{9}{2}}} - \frac{35 b e^{2} \sqrt{d + e x}}{8 \left (a + b x\right ) \left (a e - b d\right )^{4}} - \frac{35 b e \sqrt{d + e x}}{12 \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} - \frac{7 b \sqrt{d + e x}}{3 \left (a + b x\right )^{3} \left (a e - b d\right )^{2}} - \frac{2}{\left (a + b x\right )^{3} \sqrt{d + e x} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.509525, size = 141, normalized size = 0.82 \[ \frac{35 \sqrt{b} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{9/2}}-\frac{\sqrt{d+e x} \left (-\frac{22 b e (b d-a e)}{(a+b x)^2}+\frac{8 b (b d-a e)^2}{(a+b x)^3}+\frac{57 b e^2}{a+b x}+\frac{48 e^3}{d+e x}\right )}{24 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

-(Sqrt[d + e*x]*((8*b*(b*d - a*e)^2)/(a + b*x)^3 - (22*b*e*(b*d - a*e))/(a + b*x
)^2 + (57*b*e^2)/(a + b*x) + (48*e^3)/(d + e*x)))/(24*(b*d - a*e)^4) + (35*Sqrt[
b]*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*d - a*e)^(9/2))

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Maple [B]  time = 0.032, size = 292, normalized size = 1.7 \[ -2\,{\frac{{e}^{3}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-{\frac{19\,{b}^{3}{e}^{3}}{8\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{17\,{e}^{4}{b}^{2}a}{3\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{b}^{3}{e}^{3}d}{3\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{29\,{e}^{5}b{a}^{2}}{8\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{29\,{e}^{4}{b}^{2}ad}{4\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{29\,{b}^{3}{e}^{3}{d}^{2}}{8\, \left ( ae-bd \right ) ^{4} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{35\,b{e}^{3}}{8\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238401, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/48*(210*b^3*e^3*x^3 + 16*b^3*d^3 - 76*a*b^2*d^2*e + 174*a^2*b*d*e^2 + 96*a^3
*e^3 + 70*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 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(e*x + d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a
*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 14*(2*b^3*d^2
*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3)*x)/((a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b
^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*
e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^
3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 + a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*
d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*b^2*d*e^3 + a^6*b*e^4)*x)*sqrt(e*x + d)), -1/2
4*(105*b^3*e^3*x^3 + 8*b^3*d^3 - 38*a*b^2*d^2*e + 87*a^2*b*d*e^2 + 48*a^3*e^3 +
35*(b^3*d*e^2 + 8*a*b^2*e^3)*x^2 - 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(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/
(b*d - a*e))/(sqrt(e*x + d)*b)) - 7*(2*b^3*d^2*e - 14*a*b^2*d*e^2 - 33*a^2*b*e^3
)*x)/((a^3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e
^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^
4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3*d*e^3 +
a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d^2*e^2 - 4*a^5*
b^2*d*e^3 + a^6*b*e^4)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{4} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Integral(1/((a + b*x)**4*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.217122, size = 437, normalized size = 2.53 \[ -\frac{35 \, b \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \, e^{3}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{x e + d}} - \frac{57 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{3} + 87 \, \sqrt{x e + d} b^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{4} - 174 \, \sqrt{x e + d} a b^{2} d e^{4} + 87 \, \sqrt{x e + d} a^{2} b e^{5}}{24 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

-35/8*b*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^4*d^4 - 4*a*b^3*d^3
*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) - 2*e^3/
((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(x*
e + d)) - 1/24*(57*(x*e + d)^(5/2)*b^3*e^3 - 136*(x*e + d)^(3/2)*b^3*d*e^3 + 87*
sqrt(x*e + d)*b^3*d^2*e^3 + 136*(x*e + d)^(3/2)*a*b^2*e^4 - 174*sqrt(x*e + d)*a*
b^2*d*e^4 + 87*sqrt(x*e + d)*a^2*b*e^5)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^
2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((x*e + d)*b - b*d + a*e)^3)