3.2014 \(\int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=208 \[ -\frac{35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}+\frac{35 c d e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^4}+\frac{35 e^2}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{7 e}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.378809, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}+\frac{35 c d e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^4}+\frac{35 e^2}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac{7 e}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)^2} \]

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)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 80.3444, size = 187, normalized size = 0.9 \[ \frac{35 c^{\frac{3}{2}} d^{\frac{3}{2}} e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{9}{2}}} + \frac{35 c d e^{2}}{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{35 e^{2}}{12 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{7 e}{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.459227, size = 202, normalized size = 0.97 \[ \frac{-8 a^3 e^6+8 a^2 c d e^4 (10 d+7 e x)+a c^2 d^2 e^2 \left (39 d^2+238 d e x+175 e^2 x^2\right )+c^3 d^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{35 c^{3/2} d^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}} \]

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)^3,x]

[Out]

(-8*a^3*e^6 + 8*a^2*c*d*e^4*(10*d + 7*e*x) + a*c^2*d^2*e^2*(39*d^2 + 238*d*e*x +
175*e^2*x^2) + c^3*d^3*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3))/(12
*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2*(d + e*x)^(3/2)) - (35*c^(3/2)*d^(3/2)*e^2*Ar
cTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(4*(c*d^2 - a*e^2)^(
9/2))

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Maple [A]  time = 0.033, size = 262, normalized size = 1.3 \[ -{\frac{2\,{e}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{{e}^{2}cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{d}^{3}{e}^{2}{c}^{3}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,{d}^{2}{e}^{4}{c}^{2}a}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{d}^{4}{e}^{2}{c}^{3}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,{e}^{2}{c}^{2}{d}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

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)^3,x)

[Out]

-2/3*e^2/(a*e^2-c*d^2)^3/(e*x+d)^(3/2)+6*e^2/(a*e^2-c*d^2)^4*c*d/(e*x+d)^(1/2)+1
1/4*e^2/(a*e^2-c*d^2)^4*c^3*d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)+13/4*e^4/(a*e^2-
c*d^2)^4*c^2*d^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)*a-13/4*e^2/(a*e^2-c*d^2)^4*c^3*
d^4/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)+35/4*e^2/(a*e^2-c*d^2)^4*c^2*d^2/((a*e^2-c*d
^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*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(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246983, 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)^3,x, algorithm="fricas")

[Out]

[1/24*(210*c^3*d^3*e^3*x^3 - 12*c^3*d^6 + 78*a*c^2*d^4*e^2 + 160*a^2*c*d^2*e^4 -
16*a^3*e^6 + 70*(4*c^3*d^4*e^2 + 5*a*c^2*d^2*e^4)*x^2 + 105*(c^3*d^3*e^3*x^3 +
a^2*c*d^2*e^4 + (c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 + (2*a*c^2*d^3*e^3 + a^2*c*d
*e^5)*x)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2
- 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2)))/(c*d*x + a*e)) + 14
*(3*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 8*a^2*c*d*e^5)*x)/((a^2*c^4*d^9*e^2 - 4*a^3*c
^3*d^7*e^4 + 6*a^4*c^2*d^5*e^6 - 4*a^5*c*d^3*e^8 + a^6*d*e^10 + (c^6*d^10*e - 4*
a*c^5*d^8*e^3 + 6*a^2*c^4*d^6*e^5 - 4*a^3*c^3*d^4*e^7 + a^4*c^2*d^2*e^9)*x^3 + (
c^6*d^11 - 2*a*c^5*d^9*e^2 - 2*a^2*c^4*d^7*e^4 + 8*a^3*c^3*d^5*e^6 - 7*a^4*c^2*d
^3*e^8 + 2*a^5*c*d*e^10)*x^2 + (2*a*c^5*d^10*e - 7*a^2*c^4*d^8*e^3 + 8*a^3*c^3*d
^6*e^5 - 2*a^4*c^2*d^4*e^7 - 2*a^5*c*d^2*e^9 + a^6*e^11)*x)*sqrt(e*x + d)), 1/12
*(105*c^3*d^3*e^3*x^3 - 6*c^3*d^6 + 39*a*c^2*d^4*e^2 + 80*a^2*c*d^2*e^4 - 8*a^3*
e^6 + 35*(4*c^3*d^4*e^2 + 5*a*c^2*d^2*e^4)*x^2 - 105*(c^3*d^3*e^3*x^3 + a^2*c*d^
2*e^4 + (c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 + (2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x)
*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(-c*d/(c*d
^2 - a*e^2))/(sqrt(e*x + d)*c*d)) + 7*(3*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 8*a^2*c*
d*e^5)*x)/((a^2*c^4*d^9*e^2 - 4*a^3*c^3*d^7*e^4 + 6*a^4*c^2*d^5*e^6 - 4*a^5*c*d^
3*e^8 + a^6*d*e^10 + (c^6*d^10*e - 4*a*c^5*d^8*e^3 + 6*a^2*c^4*d^6*e^5 - 4*a^3*c
^3*d^4*e^7 + a^4*c^2*d^2*e^9)*x^3 + (c^6*d^11 - 2*a*c^5*d^9*e^2 - 2*a^2*c^4*d^7*
e^4 + 8*a^3*c^3*d^5*e^6 - 7*a^4*c^2*d^3*e^8 + 2*a^5*c*d*e^10)*x^2 + (2*a*c^5*d^1
0*e - 7*a^2*c^4*d^8*e^3 + 8*a^3*c^3*d^6*e^5 - 2*a^4*c^2*d^4*e^7 - 2*a^5*c*d^2*e^
9 + a^6*e^11)*x)*sqrt(e*x + d))]

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^3,x, algorithm="giac")

[Out]

Timed out