3.2065 \(\int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=257 \[ \frac{5 c d e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 e}{3 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{5 c d e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

[Out]

(-2*Sqrt[d + e*x])/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3
/2)) - (5*e)/(3*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2]) + (5*c*d*e*Sqrt[d + e*x])/((c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) + (5*c*d*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])])/(c*d^2 - a*e^2)^(7/2)

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Rubi [A]  time = 0.578928, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{5 c d e \sqrt{d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 e}{3 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{5 c d e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x])/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3
/2)) - (5*e)/(3*(c*d^2 - a*e^2)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2]) + (5*c*d*e*Sqrt[d + e*x])/((c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) + (5*c*d*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])])/(c*d^2 - a*e^2)^(7/2)

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Rubi in Sympy [A]  time = 102.097, size = 241, normalized size = 0.94 \[ \frac{5 c d e^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} - \frac{5 c d e \sqrt{d + e x}}{\left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{5 e}{3 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \sqrt{d + e x}}{3 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*c*d*e**(3/2)*atanh(sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(sqr
t(d + e*x)*sqrt(a*e**2 - c*d**2)))/(a*e**2 - c*d**2)**(7/2) - 5*c*d*e*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))) - 5*e/
(3*sqrt(d + e*x)*(a*e**2 - c*d**2)**2*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d*
*2))) + 2*sqrt(d + e*x)/(3*(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 = 0.589729, size = 185, normalized size = 0.72 \[ \frac{(d+e x)^{5/2} \left (\frac{\left (3 a^2 e^4+2 a c d e^2 (7 d+10 e x)+c^2 d^2 \left (-2 d^2+10 d e x+15 e^2 x^2\right )\right ) (a e+c d x)}{3 (d+e x) \left (c d^2-a e^2\right )^3}+\frac{5 c d e^{3/2} (a e+c d x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\left (a e^2-c d^2\right )^{7/2}}\right )}{((d+e x) (a e+c d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[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)*(3*a^2*e^4 + 2*a*c*d*e^2*(7*d + 10*e*x) + c^2*d
^2*(-2*d^2 + 10*d*e*x + 15*e^2*x^2)))/(3*(c*d^2 - a*e^2)^3*(d + e*x)) + (5*c*d*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)^(7/2)))/((a*e + c*d*x)*(d + e*x))^(5/2)

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Maple [A]  time = 0.043, size = 433, normalized size = 1.7 \[{\frac{1}{3\, \left ( cdx+ae \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}{x}^{2}{c}^{2}{d}^{2}{e}^{3}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xacd{e}^{4}\sqrt{cdx+ae}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) \sqrt{cdx+ae}x{c}^{2}{d}^{3}{e}^{2}+15\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) ac{d}^{2}{e}^{3}\sqrt{cdx+ae}-15\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{x}^{2}{c}^{2}{d}^{2}{e}^{2}-20\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}xacd{e}^{3}-10\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}x{c}^{2}{d}^{3}e-3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{a}^{2}{e}^{4}-14\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}ac{d}^{2}{e}^{2}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}{c}^{2}{d}^{4} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*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^2*d^2*e^3+15*arctanh(e*(c*d*x+a*e)^
(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c*d*e^4*(c*d*x+a*e)^(1/2)+15*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^2*d^3*e^2+15*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c*d^2*e^3*(c*d*x+a*e)^(1/2)-15*((
a*e^2-c*d^2)*e)^(1/2)*x^2*c^2*d^2*e^2-20*((a*e^2-c*d^2)*e)^(1/2)*x*a*c*d*e^3-10*
((a*e^2-c*d^2)*e)^(1/2)*x*c^2*d^3*e-3*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4-14*((a*e^2
-c*d^2)*e)^(1/2)*a*c*d^2*e^2+2*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(3/2)/(c
*d*x+a*e)^2/(a*e^2-c*d^2)^3/((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(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238929, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(15*(c^3*d^3*e^3*x^4 + a^2*c*d^3*e^3 + 2*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3
+ (c^3*d^5*e + 4*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^2 + 2*(a*c^2*d^4*e^2 + a^2*c*d^
2*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*(15*c^2*d^2*e^2*x^2
- 2*c^2*d^4 + 14*a*c*d^2*e^2 + 3*a^2*e^4 + 10*(c^2*d^3*e + 2*a*c*d*e^3)*x)*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*d^8*e^2 - 3*a^3*c
^2*d^6*e^4 + 3*a^4*c*d^4*e^6 - a^5*d^2*e^8 + (c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*
a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^4 + 2*(c^5*d^9*e - 2*a*c^4*d^7*e^3 + 2*a^3*
c^2*d^3*e^7 - a^4*c*d*e^9)*x^3 + (c^5*d^10 + a*c^4*d^8*e^2 - 8*a^2*c^3*d^6*e^4 +
8*a^3*c^2*d^4*e^6 - a^4*c*d^2*e^8 - a^5*e^10)*x^2 + 2*(a*c^4*d^9*e - 2*a^2*c^3*
d^7*e^3 + 2*a^4*c*d^3*e^7 - a^5*d*e^9)*x), -1/3*(15*(c^3*d^3*e^3*x^4 + a^2*c*d^3
*e^3 + 2*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3 + (c^3*d^5*e + 4*a*c^2*d^3*e^3 + a^2*
c*d*e^5)*x^2 + 2*(a*c^2*d^4*e^2 + a^2*c*d^2*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arct
an(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)))) - (15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 14*a*c*d^2*e^2 + 3*a^2*e^4 + 10*(c^2
*d^3*e + 2*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d))/(a^2*c^3*d^8*e^2 - 3*a^3*c^2*d^6*e^4 + 3*a^4*c*d^4*e^6 - a^5*d^2*e^8 + (c^5*
d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*x^4 + 2*(c^5*d^
9*e - 2*a*c^4*d^7*e^3 + 2*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x^3 + (c^5*d^10 + a*c^4
*d^8*e^2 - 8*a^2*c^3*d^6*e^4 + 8*a^3*c^2*d^4*e^6 - a^4*c*d^2*e^8 - a^5*e^10)*x^2
+ 2*(a*c^4*d^9*e - 2*a^2*c^3*d^7*e^3 + 2*a^4*c*d^3*e^7 - a^5*d*e^9)*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((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 = 0.656351, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x