3.455 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx\)
Optimal. Leaf size=395 \[ \frac{\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 x^4} \]
[Out]
-((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(2*a*d*e + (c*d^2 + a*
e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*a^3*d^4*e^3*x^2) - (a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*c)/(a*e) - (7*e)/d^2
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*
c*d^2*e^2 - 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*
d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*ArcTan
h[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])])/(256*a^(7/2)*d^(9/2)*e^(7/2))
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Rubi [A] time = 1.39558, antiderivative size = 395, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15
\[ \frac{\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac{\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5}-\frac{\left (\frac{3 c}{a e}-\frac{7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 x^4} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]
[Out]
-((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(2*a*d*e + (c*d^2 + a*
e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*a^3*d^4*e^3*x^2) - (a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*c)/(a*e) - (7*e)/d^2
)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*
c*d^2*e^2 - 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*
d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*ArcTan
h[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])])/(256*a^(7/2)*d^(9/2)*e^(7/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**6/(e*x+d),x)
[Out]
Timed out
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Mathematica [A] time = 0.593417, size = 407, normalized size = 1.03 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-15 x^5 \log (x) \left (c d^2-a e^2\right )^3 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right )+15 x^5 \left (c d^2-a e^2\right )^3 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^4 e^4 \left (384 d^4+48 d^3 e x-56 d^2 e^2 x^2+70 d e^3 x^3-105 e^4 x^4\right )+2 a^3 c d^2 e^3 x \left (264 d^3+48 d^2 e x-61 d e^2 x^2+95 e^3 x^3\right )+6 a^2 c^2 d^4 e^2 x^2 \left (4 d^2+3 d e x-6 e^2 x^2\right )-30 a c^3 d^6 e x^3 (d+e x)+45 c^4 d^8 x^4\right )\right )}{3840 a^{7/2} d^{9/2} e^{7/2} x^5 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]
[Out]
(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(45*c^4*d^8*x^4 - 30*a*c^3*d^6*e*x^3*(d + e*x) + 6*a^2*c^2*d^4*e^2*
x^2*(4*d^2 + 3*d*e*x - 6*e^2*x^2) + 2*a^3*c*d^2*e^3*x*(264*d^3 + 48*d^2*e*x - 61
*d*e^2*x^2 + 95*e^3*x^3) + a^4*e^4*(384*d^4 + 48*d^3*e*x - 56*d^2*e^2*x^2 + 70*d
*e^3*x^3 - 105*e^4*x^4)) - 15*(c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a
^2*e^4)*x^5*Log[x] + 15*(c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4
)*x^5*Log[c*d^2*x + 2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] +
a*e*(2*d + e*x)]))/(3840*a^(7/2)*d^(9/2)*e^(7/2)*x^5*Sqrt[(a*e + c*d*x)*(d + e*x
)])
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Maple [B] time = 0.043, size = 2888, normalized size = 7.3 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x)
[Out]
-3/128*d^4/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5-1/128*d^2/a*e/(
a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2))/x)*c^3+3/256*d^6/a^3/e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^
2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^5+1/8/d/a^2/e
^2/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+3/64*d/a^4/e^2*c^4*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+1/64*d/a^4/e^4/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(5/2)*c^3-103/192/d^4/a^2*e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/64/
e*d^2/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4+1/16/d^7*e^10*a^3/c*ln((
1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2))/(c*d*e)^(1/2)+3/64/d^2*e^3/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2+3
/16/d^3*e^6*a*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)*c-1/16/d^7*e^10*a^3/c*ln((1/2*a*e^2-1/2*c*d^
2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c
*d*e)^(1/2)-3/16/d^3*e^6*a*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2
)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+9/64/d/a^3/e^2/x^
2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+263/384/d^5/a*e^4*c*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*x+103/192/d^3/a^2*e^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(3/2)*x-1/128*d^3/a^5/e^4*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/
4/d^2/a^2/e/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/256*d^4/a^2/e/(a*d*e
)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2))/x)*c^4+15/256/d^2*a*e^5/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2
*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c-23/96/d^2/a^3/e/x*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+1/128*d^2/a^5/e^5/x*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(5/2)*c^4-1/32*d/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^
3+3/8/d^3/a/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+7/128/d^5*a*e^6*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/128/d^3*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)*c+7/48/a^3/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3-3/128*e^3/(a*d*e
)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2))/x)*c^2-1/8/d^3*e^4*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-
1/8/d^7*e^8*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/32/d*e^2/a*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-3/16/d^5*e^8*a^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*
e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-1/16/d
*e^4*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2))/(c*d*e)^(1/2)+1/4/d^6*e^7*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d
/e))^(1/2)*x+1/8/d^7*e^8*a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/1
6/d^5*e^8*a^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e
)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+1/3/d^6*e^5*(c*d*e*(x+d/e)^2+(a*
e^2-c*d^2)*(x+d/e))^(3/2)+45/128/d^6*e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
-1/4/d^6*e^7*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-1/4/d^4*e^5*c*(c*d*e*(x
+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/16/d*e^4*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x
+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e
)^(1/2)-1/32/a^2*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-3/64/a^4/e^3/x*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3-1/16/a^3/e^3/x^3*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(5/2)*c^2-1/5/d^2/a/e/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
+227/384/d^4/a*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c+19/48/d^2/a^2*e*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2-3/128*d^3/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*c^4-25/48/d^4/a*e/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
-7/256/d^4*a^2*e^7/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)-3/128*d^5/a^4/e^4*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*c^5+121/192/d^5/a*e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)-263/384/d^6/a*e^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+3/128*d^2/a^4/e
^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^4-1/128*d^4/a^5/e^5*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*c^5+23/96/d/a^3*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)*x+73/192/d^3/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+39/128/d^4*e
^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c
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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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^6),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^6),x, algorithm="fricas")
[Out]
Exception raised: TypeError
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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**6/(e*x+d),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 17.9608, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^6),x, algorithm="giac")
[Out]
Done