3.2001 \(\int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}-\frac{(d+e x)^{3/2}}{c d (a e+c d x)}+\frac{3 e \sqrt{d+e x}}{c^2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.185039, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}}-\frac{(d+e x)^{3/2}}{c d (a e+c d x)}+\frac{3 e \sqrt{d+e x}}{c^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.1522, size = 99, normalized size = 0.88 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{c d \left (a e + c d x\right )} + \frac{3 e \sqrt{d + e x}}{c^{2} d^{2}} - \frac{3 e \sqrt{a e^{2} - c d^{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.243372, size = 110, normalized size = 0.98 \[ \frac{\sqrt{d+e x} \left (3 a e^2-c d (d-2 e x)\right )}{c^2 d^2 (a e+c d x)}-\frac{3 e \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 183, normalized size = 1.6 \[ 2\,{\frac{e\sqrt{ex+d}}{{c}^{2}{d}^{2}}}+{\frac{a{e}^{3}}{{c}^{2}{d}^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-{\frac{e}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{a{e}^{3}}{{c}^{2}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+3\,{\frac{e}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224586, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d e x + a e^{2}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{2 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac{3 \,{\left (c d e x + a e^{2}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d}}{c^{3} d^{3} x + a c^{2} d^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out