3.2023 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=199 \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.371855, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 77.1693, size = 178, normalized size = 0.89 \[ \frac{c^{2} d^{2} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 e^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} - \frac{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.282346, size = 164, normalized size = 0.82 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (c^2 d^2 (d+e x)^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )+\sqrt{e} \sqrt{a e^2-c d^2} \sqrt{a e+c d x} \left (c d (d-e x)-2 a e^2\right )\right )}{4 e^{3/2} (d+e x)^{5/2} \left (a e^2-c d^2\right )^{3/2} \sqrt{a e+c d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.032, size = 292, normalized size = 1.5 \[{\frac{1}{ \left ( 4\,a{e}^{2}-4\,c{d}^{2} \right ) e}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ({\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ){x}^{2}{c}^{2}{d}^{2}{e}^{2}+2\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{2}{d}^{3}e+{\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ){c}^{2}{d}^{4}-xcde\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{e}^{2}+\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}c{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(1/2)/(e*x+d)^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229456, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}}{\left (c d e x - c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d} +{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} +{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{8 \,{\left (c d^{5} e - a d^{3} e^{3} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{3} + 3 \,{\left (c d^{3} e^{3} - a d e^{5}\right )} x^{2} + 3 \,{\left (c d^{4} e^{2} - a d^{2} e^{4}\right )} x\right )} \sqrt{-c d^{2} e + a e^{3}}}, \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}}{\left (c d e x - c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d} -{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right )}{4 \,{\left (c d^{5} e - a d^{3} e^{3} +{\left (c d^{2} e^{4} - a e^{6}\right )} x^{3} + 3 \,{\left (c d^{3} e^{3} - a d e^{5}\right )} x^{2} + 3 \,{\left (c d^{4} e^{2} - a d^{2} e^{4}\right )} x\right )} \sqrt{c d^{2} e - a e^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(7/2),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)*(c*d*
e*x - c*d^2 + 2*a*e^2)*sqrt(e*x + d) + (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)))/((c*d^5*e - a*d^3*e^3 + (c*d^2*e^
4 - a*e^6)*x^3 + 3*(c*d^3*e^3 - a*d*e^5)*x^2 + 3*(c*d^4*e^2 - a*d^2*e^4)*x)*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)*(c*d*e*x - c*d^2 + 2*a*e^2)*sqrt(e*x + d) - (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)))/((c*d^5*e - a*d^3*e^3 + (c*d^2*e^4 - a*e^6)*x^3 + 3*(c*d^3*e^3 -
a*d*e^5)*x^2 + 3*(c*d^4*e^2 - a*d^2*e^4)*x)*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)**(1/2)/(e*x+d)**(7/2),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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Timed out