3.2066 \(\int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=329 \[ \frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}-\frac{35 c d e}{12 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)
) - (7*c*d*Sqrt[d + e*x])/(6*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2)) - (35*c*d*e)/(12*(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^2*d^2*e*Sqrt[d + e*x])/(4*(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^2*d^2*e^(3/2)*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])])/(4*(c*d^2 - a*e^2)^(9/2))

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Rubi [A]  time = 0.724335, antiderivative size = 329, 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 \[ \frac{35 c^2 d^2 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{35 c^2 d^2 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}-\frac{35 c d e}{12 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{7 c d \sqrt{d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)
) - (7*c*d*Sqrt[d + e*x])/(6*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2)) - (35*c*d*e)/(12*(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^2*d^2*e*Sqrt[d + e*x])/(4*(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^2*d^2*e^(3/2)*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])])/(4*(c*d^2 - a*e^2)^(9/2))

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Rubi in Sympy [A]  time = 134.219, size = 311, normalized size = 0.95 \[ - \frac{35 c^{2} d^{2} e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{9}{2}}} + \frac{35 c^{2} d^{2} e \sqrt{d + e x}}{4 \left (a e^{2} - c d^{2}\right )^{4} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{35 c d e}{12 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{7 c d \sqrt{d + e x}}{6 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} - \frac{1}{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35*c**2*d**2*e**(3/2)*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)))/(4*(a*e**2 - c*d**2)**(9/2)) + 35*c**
2*d**2*e*sqrt(d + e*x)/(4*(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*d*e/(12*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*sqrt(d + e*x)/(6*(a*e**2 - c*d**2)**2*(
a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)) - 1/(2*sqrt(d + e*x)*(a*e**2 -
c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2))

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Mathematica [A]  time = 0.891611, size = 236, normalized size = 0.72 \[ \frac{(d+e x)^{5/2} \left (-\frac{\left (6 a^3 e^6-3 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (40 d^2+119 d e x+70 e^2 x^2\right )+c^3 d^3 \left (8 d^3-56 d^2 e x-175 d e^2 x^2-105 e^3 x^3\right )\right ) (a e+c d x)}{3 (d+e x)^2 \left (c d^2-a e^2\right )^4}-\frac{35 c^2 d^2 e^{3/2} (a e+c d x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{9/2}}\right )}{4 ((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(5/2)*(-((a*e + c*d*x)*(6*a^3*e^6 - 3*a^2*c*d*e^4*(13*d + 7*e*x) - 2*
a*c^2*d^2*e^2*(40*d^2 + 119*d*e*x + 70*e^2*x^2) + c^3*d^3*(8*d^3 - 56*d^2*e*x -
175*d*e^2*x^2 - 105*e^3*x^3)))/(3*(c*d^2 - a*e^2)^4*(d + e*x)^2) - (35*c^2*d^2*e
^(3/2)*(a*e + c*d*x)^(5/2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[-(c*d^2) + a
*e^2]])/(-(c*d^2) + a*e^2)^(9/2)))/(4*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [B]  time = 0.044, size = 668, normalized size = 2. \[ -{\frac{1}{12\, \left ( cdx+ae \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{3}{c}^{3}{d}^{3}{e}^{4}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}a{c}^{2}{d}^{2}{e}^{5}\sqrt{cdx+ae}+210\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{3}{d}^{4}{e}^{3}+210\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xa{c}^{2}{d}^{3}{e}^{4}\sqrt{cdx+ae}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{3}{d}^{5}{e}^{2}-105\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{3}{c}^{3}{d}^{3}{e}^{3}+105\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{c}^{2}{d}^{4}{e}^{3}\sqrt{cdx+ae}-140\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-175\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{3}{d}^{4}{e}^{2}-21\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{a}^{2}cd{e}^{5}-238\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xa{c}^{2}{d}^{3}{e}^{3}-56\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{3}{d}^{5}e+6\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{3}{e}^{6}-39\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}c{d}^{2}{e}^{4}-80\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}a{c}^{2}{d}^{4}{e}^{2}+8\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{3}{d}^{6} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(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+105*arctanh(e*(c*d*x+a
*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*a*c^2*d^2*e^5*(c*d*x+a*e)^(1/2)+210*arcta
nh(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+210*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^2*d^3*e^4*(c*d*
x+a*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*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))*a*c^2*d^4*e^3*(c*d*x+a*e)^(1/2)-140*(
(a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^2*d^2*e^4-175*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^3*d^4
*e^2-21*((a*e^2-c*d^2)*e)^(1/2)*x*a^2*c*d*e^5-238*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^
2*d^3*e^3-56*((a*e^2-c*d^2)*e)^(1/2)*x*c^3*d^5*e+6*((a*e^2-c*d^2)*e)^(1/2)*a^3*e
^6-39*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-80*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4
*e^2+8*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e*x+d)^(5/2)/(c*d*x+a*e)^2/(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)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24813, 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)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/24*(105*(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5*e^3 + 2*a*c^3*d^3*e^5
)*x^4 + (3*c^4*d^6*e^2 + 6*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + (c^4*d^7*e + 6
*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*x^2 + (2*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4)*
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 - 8*c^
3*d^6 + 80*a*c^2*d^4*e^2 + 39*a^2*c*d^2*e^4 - 6*a^3*e^6 + 35*(5*c^3*d^4*e^2 + 4*
a*c^2*d^2*e^4)*x^2 + 7*(8*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 3*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^2*c^4*d^11*e^2 - 4*a^3*
c^3*d^9*e^4 + 6*a^4*c^2*d^7*e^6 - 4*a^5*c*d^5*e^8 + a^6*d^3*e^10 + (c^6*d^10*e^3
- 4*a*c^5*d^8*e^5 + 6*a^2*c^4*d^6*e^7 - 4*a^3*c^3*d^4*e^9 + a^4*c^2*d^2*e^11)*x
^5 + (3*c^6*d^11*e^2 - 10*a*c^5*d^9*e^4 + 10*a^2*c^4*d^7*e^6 - 5*a^4*c^2*d^3*e^1
0 + 2*a^5*c*d*e^12)*x^4 + (3*c^6*d^12*e - 6*a*c^5*d^10*e^3 - 5*a^2*c^4*d^8*e^5 +
20*a^3*c^3*d^6*e^7 - 15*a^4*c^2*d^4*e^9 + 2*a^5*c*d^2*e^11 + a^6*e^13)*x^3 + (c
^6*d^13 + 2*a*c^5*d^11*e^2 - 15*a^2*c^4*d^9*e^4 + 20*a^3*c^3*d^7*e^6 - 5*a^4*c^2
*d^5*e^8 - 6*a^5*c*d^3*e^10 + 3*a^6*d*e^12)*x^2 + (2*a*c^5*d^12*e - 5*a^2*c^4*d^
10*e^3 + 10*a^4*c^2*d^6*e^7 - 10*a^5*c*d^4*e^9 + 3*a^6*d^2*e^11)*x), -1/12*(105*
(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5*e^3 + 2*a*c^3*d^3*e^5)*x^4 + (3*
c^4*d^6*e^2 + 6*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + (c^4*d^7*e + 6*a*c^3*d^5*
e^3 + 3*a^2*c^2*d^3*e^5)*x^2 + (2*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4)*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 - 8*c^3*d^6 + 80*a*c^2*d^4*e^
2 + 39*a^2*c*d^2*e^4 - 6*a^3*e^6 + 35*(5*c^3*d^4*e^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*
(8*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 3*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^2*c^4*d^11*e^2 - 4*a^3*c^3*d^9*e^4 + 6*a^4*c^2
*d^7*e^6 - 4*a^5*c*d^5*e^8 + a^6*d^3*e^10 + (c^6*d^10*e^3 - 4*a*c^5*d^8*e^5 + 6*
a^2*c^4*d^6*e^7 - 4*a^3*c^3*d^4*e^9 + a^4*c^2*d^2*e^11)*x^5 + (3*c^6*d^11*e^2 -
10*a*c^5*d^9*e^4 + 10*a^2*c^4*d^7*e^6 - 5*a^4*c^2*d^3*e^10 + 2*a^5*c*d*e^12)*x^4
+ (3*c^6*d^12*e - 6*a*c^5*d^10*e^3 - 5*a^2*c^4*d^8*e^5 + 20*a^3*c^3*d^6*e^7 - 1
5*a^4*c^2*d^4*e^9 + 2*a^5*c*d^2*e^11 + a^6*e^13)*x^3 + (c^6*d^13 + 2*a*c^5*d^11*
e^2 - 15*a^2*c^4*d^9*e^4 + 20*a^3*c^3*d^7*e^6 - 5*a^4*c^2*d^5*e^8 - 6*a^5*c*d^3*
e^10 + 3*a^6*d*e^12)*x^2 + (2*a*c^5*d^12*e - 5*a^2*c^4*d^10*e^3 + 10*a^4*c^2*d^6
*e^7 - 10*a^5*c*d^4*e^9 + 3*a^6*d^2*e^11)*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)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.587057, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^(5/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

sage0*x