3.1651 \(\int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=200 \[ \frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}-\frac{105 b e^3}{8 \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]

[Out]

(-35*e^3)/(8*(b*d - a*e)^4*(d + e*x)^(3/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d +
e*x)^(3/2)) + (3*e)/(4*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) - (21*e^2)/(8*
(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) - (105*b*e^3)/(8*(b*d - a*e)^5*Sqrt[d +
e*x]) + (105*b^(3/2)*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(
b*d - a*e)^(11/2))

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Rubi [A]  time = 0.417831, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{105 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{11/2}}-\frac{105 b e^3}{8 \sqrt{d+e x} (b d-a e)^5}-\frac{35 e^3}{8 (d+e x)^{3/2} (b d-a e)^4}-\frac{21 e^2}{8 (a+b x) (d+e x)^{3/2} (b d-a e)^3}+\frac{3 e}{4 (a+b x)^2 (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(-35*e^3)/(8*(b*d - a*e)^4*(d + e*x)^(3/2)) - 1/(3*(b*d - a*e)*(a + b*x)^3*(d +
e*x)^(3/2)) + (3*e)/(4*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)) - (21*e^2)/(8*
(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)) - (105*b*e^3)/(8*(b*d - a*e)^5*Sqrt[d +
e*x]) + (105*b^(3/2)*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(
b*d - a*e)^(11/2))

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Rubi in Sympy [A]  time = 101.458, size = 192, normalized size = 0.96 \[ \frac{105 b^{\frac{3}{2}} e^{3} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{11}{2}}} + \frac{105 b^{2} e^{2} \sqrt{d + e x}}{8 \left (a + b x\right ) \left (a e - b d\right )^{5}} + \frac{35 b^{2} e \sqrt{d + e x}}{4 \left (a + b x\right )^{2} \left (a e - b d\right )^{4}} + \frac{7 b^{2} \sqrt{d + e x}}{\left (a + b x\right )^{3} \left (a e - b d\right )^{3}} + \frac{6 b}{\left (a + b x\right )^{3} \sqrt{d + e x} \left (a e - b d\right )^{2}} - \frac{2}{3 \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

105*b**(3/2)*e**3*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(8*(a*e - b*d)**(1
1/2)) + 105*b**2*e**2*sqrt(d + e*x)/(8*(a + b*x)*(a*e - b*d)**5) + 35*b**2*e*sqr
t(d + e*x)/(4*(a + b*x)**2*(a*e - b*d)**4) + 7*b**2*sqrt(d + e*x)/((a + b*x)**3*
(a*e - b*d)**3) + 6*b/((a + b*x)**3*sqrt(d + e*x)*(a*e - b*d)**2) - 2/(3*(a + b*
x)**3*(d + e*x)**(3/2)*(a*e - b*d))

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Mathematica [A]  time = 0.824879, size = 167, normalized size = 0.84 \[ \frac{1}{24} \left (\frac{315 b^{3/2} e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{11/2}}+\frac{\sqrt{d+e x} \left (\frac{34 b^2 e (b d-a e)}{(a+b x)^2}-\frac{8 b^2 (b d-a e)^2}{(a+b x)^3}-\frac{123 b^2 e^2}{a+b x}+\frac{16 e^3 (a e-b d)}{(d+e x)^2}-\frac{192 b e^3}{d+e x}\right )}{(b d-a e)^5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

((Sqrt[d + e*x]*((-8*b^2*(b*d - a*e)^2)/(a + b*x)^3 + (34*b^2*e*(b*d - a*e))/(a
+ b*x)^2 - (123*b^2*e^2)/(a + b*x) + (16*e^3*(-(b*d) + a*e))/(d + e*x)^2 - (192*
b*e^3)/(d + e*x)))/(b*d - a*e)^5 + (315*b^(3/2)*e^3*ArcTanh[(Sqrt[b]*Sqrt[d + e*
x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(11/2))/24

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Maple [A]  time = 0.033, size = 319, normalized size = 1.6 \[ -{\frac{2\,{e}^{3}}{3\, \left ( ae-bd \right ) ^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+8\,{\frac{{e}^{3}b}{ \left ( ae-bd \right ) ^{5}\sqrt{ex+d}}}+{\frac{41\,{e}^{3}{b}^{4}}{8\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{e}^{4}{b}^{3}a}{3\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{35\,{b}^{4}d{e}^{3}}{3\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{55\,{e}^{5}{b}^{2}{a}^{2}}{8\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}-{\frac{55\,{e}^{4}{b}^{3}ad}{4\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{55\,{e}^{3}{b}^{4}{d}^{2}}{8\, \left ( ae-bd \right ) ^{5} \left ( bex+ae \right ) ^{3}}\sqrt{ex+d}}+{\frac{105\,{b}^{2}{e}^{3}}{8\, \left ( ae-bd \right ) ^{5}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-2/3*e^3/(a*e-b*d)^4/(e*x+d)^(3/2)+8*e^3/(a*e-b*d)^5*b/(e*x+d)^(1/2)+41/8*e^3/(a
*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)+35/3*e^4/(a*e-b*d)^5*b^3/(b*e*x+a*e)^3
*(e*x+d)^(3/2)*a-35/3*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(3/2)*d+55/8*e^5
/(a*e-b*d)^5*b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*a^2-55/4*e^4/(a*e-b*d)^5*b^3/(b*e*x
+a*e)^3*(e*x+d)^(1/2)*a*d+55/8*e^3/(a*e-b*d)^5*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*d
^2+105/8*e^3/(a*e-b*d)^5*b^2/(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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244149, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(630*b^4*e^4*x^4 + 16*b^4*d^4 - 100*a*b^3*d^3*e + 330*a^2*b^2*d^2*e^2 + 4
16*a^3*b*d*e^3 - 32*a^4*e^4 + 840*(b^4*d*e^3 + 2*a*b^3*e^4)*x^3 + 126*(b^4*d^2*e
^2 + 18*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 + 315*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^
4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*e^
3 + a^3*b*e^4)*x)*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)) - 36*(b^4*d^3*e - 10*
a*b^3*d^2*e^2 - 53*a^2*b^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^3*b^5*d^6 - 5*a^4*b^4*d^5
*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^
8*d^5*e - 5*a*b^7*d^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*
d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3
*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a
*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d
*e^5 - a^7*b*e^6)*x^2 + (3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 -
20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*x)*sqrt(e*x +
d)), -1/24*(315*b^4*e^4*x^4 + 8*b^4*d^4 - 50*a*b^3*d^3*e + 165*a^2*b^2*d^2*e^2
+ 208*a^3*b*d*e^3 - 16*a^4*e^4 + 420*(b^4*d*e^3 + 2*a*b^3*e^4)*x^3 + 63*(b^4*d^2
*e^2 + 18*a*b^3*d*e^3 + 11*a^2*b^2*e^4)*x^2 - 315*(b^4*e^4*x^4 + a^3*b*d*e^3 + (
b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*
e^3 + a^3*b*e^4)*x)*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)) - 18*(b^4*d^3*e - 10*a*b^3*d^2*e^2 - 53*a^2*b
^2*d*e^3 - 8*a^3*b*e^4)*x)/((a^3*b^5*d^6 - 5*a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2
- 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d^4*e^
2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4
+ (b^8*d^6 - 2*a*b^7*d^5*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^
4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(a*b^7*d^6 - 4*a^2*b^6*d^5
*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 +
(3*a^2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*
a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a^8*e^6)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{4} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Integral(1/((a + b*x)**4*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.222147, size = 576, normalized size = 2.88 \[ -\frac{105 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \sqrt{-b^{2} d + a b e}} - \frac{315 \,{\left (x e + d\right )}^{4} b^{4} e^{3} - 840 \,{\left (x e + d\right )}^{3} b^{4} d e^{3} + 693 \,{\left (x e + d\right )}^{2} b^{4} d^{2} e^{3} - 144 \,{\left (x e + d\right )} b^{4} d^{3} e^{3} - 16 \, b^{4} d^{4} e^{3} + 840 \,{\left (x e + d\right )}^{3} a b^{3} e^{4} - 1386 \,{\left (x e + d\right )}^{2} a b^{3} d e^{4} + 432 \,{\left (x e + d\right )} a b^{3} d^{2} e^{4} + 64 \, a b^{3} d^{3} e^{4} + 693 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{5} - 432 \,{\left (x e + d\right )} a^{2} b^{2} d e^{5} - 96 \, a^{2} b^{2} d^{2} e^{5} + 144 \,{\left (x e + d\right )} a^{3} b e^{6} + 64 \, a^{3} b d e^{6} - 16 \, a^{4} e^{7}}{24 \,{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )}{\left ({\left (x e + d\right )}^{\frac{3}{2}} b - \sqrt{x e + d} b d + \sqrt{x e + d} a e\right )}^{3}} \]

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)^(5/2)),x, algorithm="giac")

[Out]

-105/8*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^3/((b^5*d^5 - 5*a*b^4*
d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*sqrt(
-b^2*d + a*b*e)) - 1/24*(315*(x*e + d)^4*b^4*e^3 - 840*(x*e + d)^3*b^4*d*e^3 + 6
93*(x*e + d)^2*b^4*d^2*e^3 - 144*(x*e + d)*b^4*d^3*e^3 - 16*b^4*d^4*e^3 + 840*(x
*e + d)^3*a*b^3*e^4 - 1386*(x*e + d)^2*a*b^3*d*e^4 + 432*(x*e + d)*a*b^3*d^2*e^4
+ 64*a*b^3*d^3*e^4 + 693*(x*e + d)^2*a^2*b^2*e^5 - 432*(x*e + d)*a^2*b^2*d*e^5
- 96*a^2*b^2*d^2*e^5 + 144*(x*e + d)*a^3*b*e^6 + 64*a^3*b*d*e^6 - 16*a^4*e^7)/((
b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^
4 - a^5*e^5)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*b*d + sqrt(x*e + d)*a*e)^3)