3.2032 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.339898, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^2 (d+e x)^{3/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 75.5451, size = 175, normalized size = 0.95 \[ - \frac{3 c^{2} d^{2} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 e^{\frac{5}{2}} \sqrt{a e^{2} - c d^{2}}} - \frac{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)/(e*x+d)**(9/2),x)

[Out]

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

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Mathematica [A]  time = 0.339343, size = 140, normalized size = 0.76 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (-\frac{3 c^2 d^2 (d+e x)^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{\sqrt{a e^2-c d^2} \sqrt{a e+c d x}}-\sqrt{e} \left (2 a e^2+c d (3 d+5 e x)\right )\right )}{4 e^{5/2} (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

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

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Maple [A]  time = 0.035, size = 281, normalized size = 1.5 \[ -{\frac{1}{4\,{e}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{2}{d}^{2}{e}^{2}+6\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{2}{d}^{3}e+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{2}{d}^{4}+5\,xcde\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}c{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(9/2),x)

[Out]

-1/4*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*
e^2-c*d^2)*e)^(1/2))*x^2*c^2*d^2*e^2+6*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*x*c^2*d^3*e+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c
^2*d^4+5*x*c*d*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*((a*e^2-c*d^2)*e)^(
1/2)*(c*d*x+a*e)^(1/2)*a*e^2+3*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c*d^2)/
(e*x+d)^(5/2)/(c*d*x+a*e)^(1/2)/e^2/((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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231522, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}}{\left (5 \, c d e x + 3 \, c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d} - 3 \,{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} +{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{8 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}, -\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}}{\left (5 \, c d e x + 3 \, c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d} + 3 \,{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right )}{4 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )} \sqrt{c d^{2} e - a e^{3}}}\right ] \]

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)^(9/2),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2*e + a*e^3)*(5*c
*d*e*x + 3*c*d^2 + 2*a*e^2)*sqrt(e*x + d) - 3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x
^2 + 3*c^2*d^4*e*x + c^2*d^5)*log(-(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*(c*d^2*e - a*e^3)*sqrt(e*x + d) + (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2
)*sqrt(-c*d^2*e + a*e^3))/(e^2*x^2 + 2*d*e*x + d^2)))/((e^5*x^3 + 3*d*e^4*x^2 +
3*d^2*e^3*x + d^3*e^2)*sqrt(-c*d^2*e + a*e^3)), -1/4*(sqrt(c*d*e*x^2 + a*d*e + (
c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*(5*c*d*e*x + 3*c*d^2 + 2*a*e^2)*sqrt(e*x
+ d) + 3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*arctan
(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)
/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)))/((e^5*x^3 + 3*d*e^4*x^2 + 3*d^2
*e^3*x + d^3*e^2)*sqrt(c*d^2*e - a*e^3))]

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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)/(e*x+d)**(9/2),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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Timed out