3.1990 \(\int \frac{(d+e x)^{7/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac{2 (d+e x)^{5/2}}{5 c d} \]

[Out]

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

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Rubi [A]  time = 0.259225, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac{2 (d+e x)^{5/2}}{5 c d} \]

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),x]

[Out]

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

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Rubi in Sympy [A]  time = 61.7603, size = 131, normalized size = 0.89 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{5 c d} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 c^{2} d^{2}} + \frac{2 \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{c^{3} d^{3}} - \frac{2 \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{7}{2}} d^{\frac{7}{2}}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.16383, size = 135, normalized size = 0.92 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^4-5 a c d e^2 (7 d+e x)+c^2 d^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )}{15 c^3 d^3}-\frac{2 \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}} \]

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),x]

[Out]

(2*Sqrt[d + e*x]*(15*a^2*e^4 - 5*a*c*d*e^2*(7*d + e*x) + c^2*d^2*(23*d^2 + 11*d*
e*x + 3*e^2*x^2)))/(15*c^3*d^3) - (2*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt
[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(7/2)*d^(7/2))

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Maple [B]  time = 0.011, size = 324, normalized size = 2.2 \[{\frac{2}{5\,cd} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,a{e}^{2}}{3\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}{e}^{4}\sqrt{ex+d}}{{c}^{3}{d}^{3}}}-4\,{\frac{a{e}^{2}\sqrt{ex+d}}{{c}^{2}d}}+2\,{\frac{d\sqrt{ex+d}}{c}}-2\,{\frac{{a}^{3}{e}^{6}}{{c}^{3}{d}^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+6\,{\frac{{a}^{2}{e}^{4}}{{c}^{2}d\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-6\,{\frac{ad{e}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251019, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 23 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} +{\left (11 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \, c^{3} d^{3}}, -\frac{2 \,{\left (15 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (3 \, c^{2} d^{2} e^{2} x^{2} + 23 \, c^{2} d^{4} - 35 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} +{\left (11 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, c^{3} d^{3}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/15*(15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*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*(3*c^2*d^2*e^2*x^2 + 23*c^2*d^4 - 35*a*c*d^2*e^2 + 15*a^2*e^4 + (
11*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^3*d^3), -2/15*(15*(c^2*d^4 - 2*
a*c*d^2*e^2 + a^2*e^4)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sqrt(-(
c*d^2 - a*e^2)/(c*d))) - (3*c^2*d^2*e^2*x^2 + 23*c^2*d^4 - 35*a*c*d^2*e^2 + 15*a
^2*e^4 + (11*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^3*d^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((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**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),x, algorithm="giac")

[Out]

Timed out