3.2060 \(\int \frac{1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=331 \[ -\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac{35 c^2 d^2}{24 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
) + (7*c*d)/(12*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (35*c^2*d^2)/(24*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[d + e*x])/(8*(c*d^2 - a*e^2)^4
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[e]*ArcTan[(Sqrt
[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e
*x])])/(8*(c*d^2 - a*e^2)^(9/2))

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Rubi [A]  time = 0.661585, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{35 c^3 d^3 \sqrt{d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{35 c^3 d^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac{35 c^2 d^2}{24 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
) + (7*c*d)/(12*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2]) + (35*c^2*d^2)/(24*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[d + e*x])/(8*(c*d^2 - a*e^2)^4
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[e]*ArcTan[(Sqrt
[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e
*x])])/(8*(c*d^2 - a*e^2)^(9/2))

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Rubi in Sympy [A]  time = 133.702, size = 311, normalized size = 0.94 \[ \frac{35 c^{3} d^{3} \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{8 \left (a e^{2} - c d^{2}\right )^{\frac{9}{2}}} - \frac{35 c^{3} d^{3} \sqrt{d + e x}}{8 \left (a e^{2} - c d^{2}\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{35 c^{2} d^{2}}{24 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{7 c d}{12 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{1}{3 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

35*c**3*d**3*sqrt(e)*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)
)/(sqrt(d + e*x)*sqrt(a*e**2 - c*d**2)))/(8*(a*e**2 - c*d**2)**(9/2)) - 35*c**3*
d**3*sqrt(d + e*x)/(8*(a*e**2 - c*d**2)**4*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))) - 35*c**2*d**2/(24*sqrt(d + e*x)*(a*e**2 - c*d**2)**3*sqrt(a*d*e + c*
d*e*x**2 + x*(a*e**2 + c*d**2))) + 7*c*d/(12*(d + e*x)**(3/2)*(a*e**2 - c*d**2)*
*2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - 1/(3*(d + e*x)**(5/2)*(a*e*
*2 - c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))

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Mathematica [A]  time = 0.856882, size = 235, normalized size = 0.71 \[ \frac{(d+e x)^{3/2} \left (\frac{35 c^3 d^3 \sqrt{e} (a e+c d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{9/2}}-\frac{(a e+c d x) \left (8 a^3 e^6-2 a^2 c d e^4 (19 d+7 e x)+a c^2 d^2 e^2 \left (87 d^2+98 d e x+35 e^2 x^2\right )+c^3 d^3 \left (48 d^3+231 d^2 e x+280 d e^2 x^2+105 e^3 x^3\right )\right )}{3 (d+e x)^3 \left (c d^2-a e^2\right )^4}\right )}{8 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

((d + e*x)^(3/2)*(-((a*e + c*d*x)*(8*a^3*e^6 - 2*a^2*c*d*e^4*(19*d + 7*e*x) + a*
c^2*d^2*e^2*(87*d^2 + 98*d*e*x + 35*e^2*x^2) + c^3*d^3*(48*d^3 + 231*d^2*e*x + 2
80*d*e^2*x^2 + 105*e^3*x^3)))/(3*(c*d^2 - a*e^2)^4*(d + e*x)^3) + (35*c^3*d^3*Sq
rt[e]*(a*e + c*d*x)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a*
e^2]])/(-(c*d^2) + a*e^2)^(9/2)))/(8*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.044, size = 559, normalized size = 1.7 \[{\frac{1}{ \left ( 24\,cdx+24\,ae \right ) \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{3}{c}^{3}{d}^{3}{e}^{4}+315\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{3}{d}^{4}{e}^{3}+315\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{3}{d}^{5}{e}^{2}-105\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{3}{c}^{3}{d}^{3}{e}^{3}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{c}^{3}{d}^{6}e-35\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-280\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{3}{d}^{4}{e}^{2}+14\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{a}^{2}cd{e}^{5}-98\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xa{c}^{2}{d}^{3}{e}^{3}-231\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{3}{d}^{5}e-8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{3}{e}^{6}+38\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}c{d}^{2}{e}^{4}-87\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{c}^{2}{d}^{4}{e}^{2}-48\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(105*arctanh(e*(c*d*x+a*e)^(1/2)/((
a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^3*c^3*d^3*e^4+315*arctanh(e*(c*d*x+a*
e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x^2*c^3*d^4*e^3+315*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*x*c^3*d^5*e^2-105
*((a*e^2-c*d^2)*e)^(1/2)*x^3*c^3*d^3*e^3+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2
-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^6*e-35*((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c
^2*d^2*e^4-280*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^3*d^4*e^2+14*((a*e^2-c*d^2)*e)^(1/2
)*x*a^2*c*d*e^5-98*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^2*d^3*e^3-231*((a*e^2-c*d^2)*e)
^(1/2)*x*c^3*d^5*e-8*((a*e^2-c*d^2)*e)^(1/2)*a^3*e^6+38*((a*e^2-c*d^2)*e)^(1/2)*
a^2*c*d^2*e^4-87*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4*e^2-48*((a*e^2-c*d^2)*e)^(1/2
)*c^3*d^6)/(e*x+d)^(7/2)/(c*d*x+a*e)/(a*e^2-c*d^2)^4/((a*e^2-c*d^2)*e)^(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/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242921, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/48*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^4
+ 2*(3*c^4*d^6*e^2 + 2*a*c^3*d^4*e^4)*x^3 + 2*(2*c^4*d^7*e + 3*a*c^3*d^5*e^3)*x^
2 + (c^4*d^8 + 4*a*c^3*d^6*e^2)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 +
2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c
*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2))
- 2*(105*c^3*d^3*e^3*x^3 + 48*c^3*d^6 + 87*a*c^2*d^4*e^2 - 38*a^2*c*d^2*e^4 + 8
*a^3*e^6 + 35*(8*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 7*(33*c^3*d^5*e + 14*a*c^2*d
^3*e^3 - 2*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x
+ d))/(a*c^4*d^12*e - 4*a^2*c^3*d^10*e^3 + 6*a^3*c^2*d^8*e^5 - 4*a^4*c*d^6*e^7 +
a^5*d^4*e^9 + (c^5*d^9*e^4 - 4*a*c^4*d^7*e^6 + 6*a^2*c^3*d^5*e^8 - 4*a^3*c^2*d^
3*e^10 + a^4*c*d*e^12)*x^5 + (4*c^5*d^10*e^3 - 15*a*c^4*d^8*e^5 + 20*a^2*c^3*d^6
*e^7 - 10*a^3*c^2*d^4*e^9 + a^5*e^13)*x^4 + 2*(3*c^5*d^11*e^2 - 10*a*c^4*d^9*e^4
+ 10*a^2*c^3*d^7*e^6 - 5*a^4*c*d^3*e^10 + 2*a^5*d*e^12)*x^3 + 2*(2*c^5*d^12*e -
5*a*c^4*d^10*e^3 + 10*a^3*c^2*d^6*e^7 - 10*a^4*c*d^4*e^9 + 3*a^5*d^2*e^11)*x^2
+ (c^5*d^13 - 10*a^2*c^3*d^9*e^4 + 20*a^3*c^2*d^7*e^6 - 15*a^4*c*d^5*e^8 + 4*a^5
*d^3*e^10)*x), 1/24*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3
*d^3*e^5)*x^4 + 2*(3*c^4*d^6*e^2 + 2*a*c^3*d^4*e^4)*x^3 + 2*(2*c^4*d^7*e + 3*a*c
^3*d^5*e^3)*x^2 + (c^4*d^8 + 4*a*c^3*d^6*e^2)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(
sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d^2 - a*e^2
)))) - (105*c^3*d^3*e^3*x^3 + 48*c^3*d^6 + 87*a*c^2*d^4*e^2 - 38*a^2*c*d^2*e^4 +
8*a^3*e^6 + 35*(8*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 7*(33*c^3*d^5*e + 14*a*c^2
*d^3*e^3 - 2*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*
x + d))/(a*c^4*d^12*e - 4*a^2*c^3*d^10*e^3 + 6*a^3*c^2*d^8*e^5 - 4*a^4*c*d^6*e^7
+ a^5*d^4*e^9 + (c^5*d^9*e^4 - 4*a*c^4*d^7*e^6 + 6*a^2*c^3*d^5*e^8 - 4*a^3*c^2*
d^3*e^10 + a^4*c*d*e^12)*x^5 + (4*c^5*d^10*e^3 - 15*a*c^4*d^8*e^5 + 20*a^2*c^3*d
^6*e^7 - 10*a^3*c^2*d^4*e^9 + a^5*e^13)*x^4 + 2*(3*c^5*d^11*e^2 - 10*a*c^4*d^9*e
^4 + 10*a^2*c^3*d^7*e^6 - 5*a^4*c*d^3*e^10 + 2*a^5*d*e^12)*x^3 + 2*(2*c^5*d^12*e
- 5*a*c^4*d^10*e^3 + 10*a^3*c^2*d^6*e^7 - 10*a^4*c*d^4*e^9 + 3*a^5*d^2*e^11)*x^
2 + (c^5*d^13 - 10*a^2*c^3*d^9*e^4 + 20*a^3*c^2*d^7*e^6 - 15*a^4*c*d^5*e^8 + 4*a
^5*d^3*e^10)*x)]

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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(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 2]