3.2015 \(\int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=244 \[ -\frac{63 c^{5/2} d^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}}+\frac{63 c^2 d^2 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^5}+\frac{21 c d e^2}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^4}+\frac{9 e}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{63 e^2}{20 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3} \]

[Out]

(63*e^2)/(20*(c*d^2 - a*e^2)^3*(d + e*x)^(5/2)) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*
d*x)^2*(d + e*x)^(5/2)) + (9*e)/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*(d + e*x)^(5/
2)) + (21*c*d*e^2)/(4*(c*d^2 - a*e^2)^4*(d + e*x)^(3/2)) + (63*c^2*d^2*e^2)/(4*(
c*d^2 - a*e^2)^5*Sqrt[d + e*x]) - (63*c^(5/2)*d^(5/2)*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)^(11/2))

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Rubi [A]  time = 0.638587, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{63 c^{5/2} d^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}}+\frac{63 c^2 d^2 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^5}+\frac{21 c d e^2}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^4}+\frac{9 e}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{63 e^2}{20 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3} \]

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)^3),x]

[Out]

(63*e^2)/(20*(c*d^2 - a*e^2)^3*(d + e*x)^(5/2)) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*
d*x)^2*(d + e*x)^(5/2)) + (9*e)/(4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)*(d + e*x)^(5/
2)) + (21*c*d*e^2)/(4*(c*d^2 - a*e^2)^4*(d + e*x)^(3/2)) + (63*c^2*d^2*e^2)/(4*(
c*d^2 - a*e^2)^5*Sqrt[d + e*x]) - (63*c^(5/2)*d^(5/2)*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)^(11/2))

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Rubi in Sympy [A]  time = 102.311, size = 221, normalized size = 0.91 \[ - \frac{63 c^{\frac{5}{2}} d^{\frac{5}{2}} e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{11}{2}}} - \frac{63 c^{2} d^{2} e^{2}}{4 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{5}} + \frac{21 c d e^{2}}{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{4}} - \frac{63 e^{2}}{20 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{3}} + \frac{9 e}{4 \left (d + e x\right )^{\frac{5}{2}} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-63*c**(5/2)*d**(5/2)*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)**(11/2)) - 63*c**2*d**2*e**2/(4*sqrt(d + e*x)*(a*e**2
- c*d**2)**5) + 21*c*d*e**2/(4*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**4) - 63*e**2/
(20*(d + e*x)**(5/2)*(a*e**2 - c*d**2)**3) + 9*e/(4*(d + e*x)**(5/2)*(a*e + c*d*
x)*(a*e**2 - c*d**2)**2) + 1/(2*(d + e*x)**(5/2)*(a*e + c*d*x)**2*(a*e**2 - c*d*
*2))

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Mathematica [A]  time = 1.04528, size = 216, normalized size = 0.89 \[ \frac{\sqrt{d+e x} \left (-\frac{75 c^3 d^3 e}{a e+c d x}+\frac{10 c^3 d^3 \left (c d^2-a e^2\right )}{(a e+c d x)^2}-\frac{8 \left (c d^2 e-a e^3\right )^2}{(d+e x)^3}+\frac{40 c d e^2 \left (a e^2-c d^2\right )}{(d+e x)^2}-\frac{240 c^2 d^2 e^2}{d+e x}\right )}{20 \left (a e^2-c d^2\right )^5}-\frac{63 c^{5/2} d^{5/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}} \]

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)^3),x]

[Out]

(Sqrt[d + e*x]*((10*c^3*d^3*(c*d^2 - a*e^2))/(a*e + c*d*x)^2 - (75*c^3*d^3*e)/(a
*e + c*d*x) - (8*(c*d^2*e - a*e^3)^2)/(d + e*x)^3 + (40*c*d*e^2*(-(c*d^2) + a*e^
2))/(d + e*x)^2 - (240*c^2*d^2*e^2)/(d + e*x)))/(20*(-(c*d^2) + a*e^2)^5) - (63*
c^(5/2)*d^(5/2)*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)^(11/2))

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Maple [A]  time = 0.031, size = 294, normalized size = 1.2 \[ -{\frac{2\,{e}^{2}}{5\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-12\,{\frac{{e}^{2}{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}\sqrt{ex+d}}}+2\,{\frac{{e}^{2}cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{3/2}}}-{\frac{15\,{d}^{4}{e}^{2}{c}^{4}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{17\,{d}^{3}{e}^{4}{c}^{3}a}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{17\,{d}^{5}{e}^{2}{c}^{4}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{63\,{e}^{2}{c}^{3}{d}^{3}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*e^2/(a*e^2-c*d^2)^3/(e*x+d)^(5/2)-12*e^2/(a*e^2-c*d^2)^5*c^2*d^2/(e*x+d)^(1
/2)+2*e^2/(a*e^2-c*d^2)^4*c*d/(e*x+d)^(3/2)-15/4*e^2/(a*e^2-c*d^2)^5*c^4*d^4/(c*
d*e*x+a*e^2)^2*(e*x+d)^(3/2)-17/4*e^4/(a*e^2-c*d^2)^5*c^3*d^3/(c*d*e*x+a*e^2)^2*
(e*x+d)^(1/2)*a+17/4*e^2/(a*e^2-c*d^2)^5*c^4*d^5/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)
-63/4*e^2/(a*e^2-c*d^2)^5*c^3*d^3/((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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266621, 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*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/40*(630*c^4*d^4*e^4*x^4 - 20*c^4*d^8 + 170*a*c^3*d^6*e^2 + 576*a^2*c^2*d^4*e^
4 - 112*a^3*c*d^2*e^6 + 16*a^4*e^8 + 210*(7*c^4*d^5*e^3 + 5*a*c^3*d^3*e^5)*x^3 +
42*(23*c^4*d^6*e^2 + 59*a*c^3*d^4*e^4 + 8*a^2*c^2*d^2*e^6)*x^2 + 315*(c^4*d^4*e
^4*x^4 + a^2*c^2*d^4*e^4 + 2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d^6*e^2 +
4*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 + a^2*c^2*d^3*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)) + 6*(15*c^4*d
^7*e + 277*a*c^3*d^5*e^3 + 136*a^2*c^2*d^3*e^5 - 8*a^3*c*d*e^7)*x)/((a^2*c^5*d^1
2*e^2 - 5*a^3*c^4*d^10*e^4 + 10*a^4*c^3*d^8*e^6 - 10*a^5*c^2*d^6*e^8 + 5*a^6*c*d
^4*e^10 - a^7*d^2*e^12 + (c^7*d^12*e^2 - 5*a*c^6*d^10*e^4 + 10*a^2*c^5*d^8*e^6 -
10*a^3*c^4*d^6*e^8 + 5*a^4*c^3*d^4*e^10 - a^5*c^2*d^2*e^12)*x^4 + 2*(c^7*d^13*e
- 4*a*c^6*d^11*e^3 + 5*a^2*c^5*d^9*e^5 - 5*a^4*c^3*d^5*e^9 + 4*a^5*c^2*d^3*e^11
- a^6*c*d*e^13)*x^3 + (c^7*d^14 - a*c^6*d^12*e^2 - 9*a^2*c^5*d^10*e^4 + 25*a^3*
c^4*d^8*e^6 - 25*a^4*c^3*d^6*e^8 + 9*a^5*c^2*d^4*e^10 + a^6*c*d^2*e^12 - a^7*e^1
4)*x^2 + 2*(a*c^6*d^13*e - 4*a^2*c^5*d^11*e^3 + 5*a^3*c^4*d^9*e^5 - 5*a^5*c^2*d^
5*e^9 + 4*a^6*c*d^3*e^11 - a^7*d*e^13)*x)*sqrt(e*x + d)), 1/20*(315*c^4*d^4*e^4*
x^4 - 10*c^4*d^8 + 85*a*c^3*d^6*e^2 + 288*a^2*c^2*d^4*e^4 - 56*a^3*c*d^2*e^6 + 8
*a^4*e^8 + 105*(7*c^4*d^5*e^3 + 5*a*c^3*d^3*e^5)*x^3 + 21*(23*c^4*d^6*e^2 + 59*a
*c^3*d^4*e^4 + 8*a^2*c^2*d^2*e^6)*x^2 - 315*(c^4*d^4*e^4*x^4 + a^2*c^2*d^4*e^4 +
2*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + (c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 + a^2*c^2*
d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 + a^2*c^2*d^3*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)) + 3*(15*c^4*d^7*e + 277*a*c^3*d^5*e^3 + 136*a^2*c^2*d^3*e^5 - 8*a^3*c*d*e
^7)*x)/((a^2*c^5*d^12*e^2 - 5*a^3*c^4*d^10*e^4 + 10*a^4*c^3*d^8*e^6 - 10*a^5*c^2
*d^6*e^8 + 5*a^6*c*d^4*e^10 - a^7*d^2*e^12 + (c^7*d^12*e^2 - 5*a*c^6*d^10*e^4 +
10*a^2*c^5*d^8*e^6 - 10*a^3*c^4*d^6*e^8 + 5*a^4*c^3*d^4*e^10 - a^5*c^2*d^2*e^12)
*x^4 + 2*(c^7*d^13*e - 4*a*c^6*d^11*e^3 + 5*a^2*c^5*d^9*e^5 - 5*a^4*c^3*d^5*e^9
+ 4*a^5*c^2*d^3*e^11 - a^6*c*d*e^13)*x^3 + (c^7*d^14 - a*c^6*d^12*e^2 - 9*a^2*c^
5*d^10*e^4 + 25*a^3*c^4*d^8*e^6 - 25*a^4*c^3*d^6*e^8 + 9*a^5*c^2*d^4*e^10 + a^6*
c*d^2*e^12 - a^7*e^14)*x^2 + 2*(a*c^6*d^13*e - 4*a^2*c^5*d^11*e^3 + 5*a^3*c^4*d^
9*e^5 - 5*a^5*c^2*d^5*e^9 + 4*a^6*c*d^3*e^11 - a^7*d*e^13)*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(1/(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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out