3.2031 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 \sqrt{d+e x}}-\frac{3 c d \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.3508, antiderivative size = 175, 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 \[ -\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 \sqrt{d+e x}}-\frac{3 c d \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{5/2}} \]

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)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 73.2124, size = 165, normalized size = 0.94 \[ \frac{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{2} \sqrt{d + e x}} - \frac{3 c d \sqrt{a e^{2} - c d^{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{e^{\frac{5}{2}}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2)/(e*x+d)**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.267675, size = 142, normalized size = 0.81 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\sqrt{e} \sqrt{a e+c d x} \left (c d (3 d+2 e x)-a e^2\right )-3 c d (d+e x) \sqrt{a e^2-c d^2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{e^{5/2} (d+e x)^{3/2} \sqrt{a e+c d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

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

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Maple [B]  time = 0.034, size = 314, normalized size = 1.8 \[{\frac{1}{{e}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( -3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) xacd{e}^{3}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{2}{d}^{3}e-3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) ac{d}^{2}{e}^{2}+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{2}{d}^{4}+2\,xcde\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}-\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{e}^{2}+3\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}c{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

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)^(7/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236194, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c^{2} d^{2} e x^{2} + 6 \, a c d^{2} e - 2 \, a^{2} e^{3} + 3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c d \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (3 \, c^{2} d^{3} + a c d e^{2}\right )} x}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{2}}, \frac{2 \, c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - a^{2} e^{3} - 3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}\right ) +{\left (3 \, c^{2} d^{3} + a c d e^{2}\right )} x}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e^{2}}\right ] \]

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)^(7/2),x, algorithm="fricas")

[Out]

[1/2*(4*c^2*d^2*e*x^2 + 6*a*c*d^2*e - 2*a^2*e^3 + 3*sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/e)*log(-(c*d*e^2*x^2 + 2
*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(e*x + d)*e*sqrt(-(c*d^2 - a*e^2)/e))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^2*d^3
+ a*c*d*e^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e^2)
, (2*c^2*d^2*e*x^2 + 3*a*c*d^2*e - a^2*e^3 - 3*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/e)*arctan(-sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)/((c*d*e^2*x^2 + a*d*e^2 +
(c*d^2*e + a*e^3)*x)*sqrt((c*d^2 - a*e^2)/e))) + (3*c^2*d^3 + a*c*d*e^2)*x)/(sq
rt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e^2)]

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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)**(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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Timed out