3.2006 \(\int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=192 \[ \frac{7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}-\frac{7 c^2 d^2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^4}-\frac{7 c d e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{7 e}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2} \]

[Out]

(-7*e)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*
(d + e*x)^(5/2)) - (7*c*d*e)/(3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) - (7*c^2*d^2*
e)/((c*d^2 - a*e^2)^4*Sqrt[d + e*x]) + (7*c^(5/2)*d^(5/2)*e*ArcTanh[(Sqrt[c]*Sqr
t[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Rubi [A]  time = 0.385163, antiderivative size = 192, 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 \[ \frac{7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}-\frac{7 c^2 d^2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^4}-\frac{7 c d e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{7 e}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-7*e)/(5*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*
(d + e*x)^(5/2)) - (7*c*d*e)/(3*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2)) - (7*c^2*d^2*
e)/((c*d^2 - a*e^2)^4*Sqrt[d + e*x]) + (7*c^(5/2)*d^(5/2)*e*ArcTanh[(Sqrt[c]*Sqr
t[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Rubi in Sympy [A]  time = 79.1758, size = 173, normalized size = 0.9 \[ - \frac{7 c^{\frac{5}{2}} d^{\frac{5}{2}} e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{9}{2}}} - \frac{7 c^{2} d^{2} e}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{4}} + \frac{7 c d e}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{3}} - \frac{7 e}{5 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{\left (d + e x\right )^{\frac{5}{2}} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*c**(5/2)*d**(5/2)*e*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))
/(a*e**2 - c*d**2)**(9/2) - 7*c**2*d**2*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)**4) +
7*c*d*e/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**3) - 7*e/(5*(d + e*x)**(5/2)*(a*
e**2 - c*d**2)**2) + 1/((d + e*x)**(5/2)*(a*e + c*d*x)*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.797419, size = 177, normalized size = 0.92 \[ \frac{7 c^{5/2} d^{5/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}-\frac{\sqrt{d+e x} \left (\frac{15 c^3 d^3}{a e+c d x}+\frac{20 c d e \left (c d^2-a e^2\right )}{(d+e x)^2}+\frac{6 e \left (c d^2-a e^2\right )^2}{(d+e x)^3}+\frac{90 c^2 d^2 e}{d+e x}\right )}{15 \left (c d^2-a e^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*((15*c^3*d^3)/(a*e + c*d*x) + (6*e*(c*d^2 - a*e^2)^2)/(d + e*x)^
3 + (20*c*d*e*(c*d^2 - a*e^2))/(d + e*x)^2 + (90*c^2*d^2*e)/(d + e*x)))/(15*(c*d
^2 - a*e^2)^4) + (7*c^(5/2)*d^(5/2)*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sq
rt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)

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Maple [A]  time = 0.028, size = 193, normalized size = 1. \[ -{\frac{2\,e}{5\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-6\,{\frac{e{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}\sqrt{ex+d}}}+{\frac{4\,dec}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{{c}^{3}e{d}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-7\,{\frac{{c}^{3}e{d}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*e/(a*e^2-c*d^2)^2/(e*x+d)^(5/2)-6*e/(a*e^2-c*d^2)^4*c^2*d^2/(e*x+d)^(1/2)+4
/3*e/(a*e^2-c*d^2)^3*c*d/(e*x+d)^(3/2)-e*c^3*d^3/(a*e^2-c*d^2)^4*(e*x+d)^(1/2)/(
c*d*e*x+a*e^2)-7*e*c^3*d^3/(a*e^2-c*d^2)^4/((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)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234642, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/30*(210*c^3*d^3*e^3*x^3 + 30*c^3*d^6 + 232*a*c^2*d^4*e^2 - 64*a^2*c*d^2*e^4
+ 12*a^3*e^6 + 70*(7*c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 - 105*(c^3*d^3*e^3*x^3 +
a*c^2*d^4*e^2 + (2*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + (c^3*d^5*e + 2*a*c^2*d^3*
e^3)*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*
(23*c^3*d^5*e + 24*a*c^2*d^3*e^3 - 2*a^2*c*d*e^5)*x)/((a*c^4*d^10*e - 4*a^2*c^3*
d^8*e^3 + 6*a^3*c^2*d^6*e^5 - 4*a^4*c*d^4*e^7 + a^5*d^2*e^9 + (c^5*d^9*e^2 - 4*a
*c^4*d^7*e^4 + 6*a^2*c^3*d^5*e^6 - 4*a^3*c^2*d^3*e^8 + a^4*c*d*e^10)*x^3 + (2*c^
5*d^10*e - 7*a*c^4*d^8*e^3 + 8*a^2*c^3*d^6*e^5 - 2*a^3*c^2*d^4*e^7 - 2*a^4*c*d^2
*e^9 + a^5*e^11)*x^2 + (c^5*d^11 - 2*a*c^4*d^9*e^2 - 2*a^2*c^3*d^7*e^4 + 8*a^3*c
^2*d^5*e^6 - 7*a^4*c*d^3*e^8 + 2*a^5*d*e^10)*x)*sqrt(e*x + d)), -1/15*(105*c^3*d
^3*e^3*x^3 + 15*c^3*d^6 + 116*a*c^2*d^4*e^2 - 32*a^2*c*d^2*e^4 + 6*a^3*e^6 + 35*
(7*c^3*d^4*e^2 + 2*a*c^2*d^2*e^4)*x^2 - 105*(c^3*d^3*e^3*x^3 + a*c^2*d^4*e^2 + (
2*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + (c^3*d^5*e + 2*a*c^2*d^3*e^3)*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*(23*c^3*d^5*e + 24*a*c^2*d^3*e^3 - 2*a^2*c*d*e^5)*x)/
((a*c^4*d^10*e - 4*a^2*c^3*d^8*e^3 + 6*a^3*c^2*d^6*e^5 - 4*a^4*c*d^4*e^7 + a^5*d
^2*e^9 + (c^5*d^9*e^2 - 4*a*c^4*d^7*e^4 + 6*a^2*c^3*d^5*e^6 - 4*a^3*c^2*d^3*e^8
+ a^4*c*d*e^10)*x^3 + (2*c^5*d^10*e - 7*a*c^4*d^8*e^3 + 8*a^2*c^3*d^6*e^5 - 2*a^
3*c^2*d^4*e^7 - 2*a^4*c*d^2*e^9 + a^5*e^11)*x^2 + (c^5*d^11 - 2*a*c^4*d^9*e^2 -
2*a^2*c^3*d^7*e^4 + 8*a^3*c^2*d^5*e^6 - 7*a^4*c*d^3*e^8 + 2*a^5*d*e^10)*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)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,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)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out