3.1999 \(\int \frac{(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=178 \[ -\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{7 e \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac{(d+e x)^{7/2}}{c d (a e+c d x)}+\frac{7 e (d+e x)^{5/2}}{5 c^2 d^2} \]

[Out]

(7*e*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(c^4*d^4) + (7*e*(c*d^2 - a*e^2)*(d + e*x)
^(3/2))/(3*c^3*d^3) + (7*e*(d + e*x)^(5/2))/(5*c^2*d^2) - (d + e*x)^(7/2)/(c*d*(
a*e + c*d*x)) - (7*e*(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^(9/2)*d^(9/2))

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Rubi [A]  time = 0.334248, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac{7 e \sqrt{d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac{(d+e x)^{7/2}}{c d (a e+c d x)}+\frac{7 e (d+e x)^{5/2}}{5 c^2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(7*e*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(c^4*d^4) + (7*e*(c*d^2 - a*e^2)*(d + e*x)
^(3/2))/(3*c^3*d^3) + (7*e*(d + e*x)^(5/2))/(5*c^2*d^2) - (d + e*x)^(7/2)/(c*d*(
a*e + c*d*x)) - (7*e*(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^(9/2)*d^(9/2))

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Rubi in Sympy [A]  time = 75.6733, size = 162, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{7}{2}}}{c d \left (a e + c d x\right )} + \frac{7 e \left (d + e x\right )^{\frac{5}{2}}}{5 c^{2} d^{2}} - \frac{7 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )}{3 c^{3} d^{3}} + \frac{7 e \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}}{c^{4} d^{4}} - \frac{7 e \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{9}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(11/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-(d + e*x)**(7/2)/(c*d*(a*e + c*d*x)) + 7*e*(d + e*x)**(5/2)/(5*c**2*d**2) - 7*e
*(d + e*x)**(3/2)*(a*e**2 - c*d**2)/(3*c**3*d**3) + 7*e*sqrt(d + e*x)*(a*e**2 -
c*d**2)**2/(c**4*d**4) - 7*e*(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**(9/2)*d**(9/2))

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Mathematica [A]  time = 0.34401, size = 191, normalized size = 1.07 \[ \frac{\sqrt{d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (7 d-2 e x)+7 a c^2 d^2 e^2 \left (23 d^2-24 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )}{15 c^4 d^4 (a e+c d x)}-\frac{7 e \left (c d^2-a e^2\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(11/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(Sqrt[d + e*x]*(105*a^3*e^6 - 35*a^2*c*d*e^4*(7*d - 2*e*x) + 7*a*c^2*d^2*e^2*(23
*d^2 - 24*d*e*x - 2*e^2*x^2) + c^3*d^3*(-15*d^3 + 116*d^2*e*x + 32*d*e^2*x^2 + 6
*e^3*x^3)))/(15*c^4*d^4*(a*e + c*d*x)) - (7*e*(c*d^2 - a*e^2)^(5/2)*ArcTanh[(Sqr
t[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(9/2)*d^(9/2))

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Maple [B]  time = 0.023, size = 457, normalized size = 2.6 \[{\frac{2\,e}{5\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{4\,a{e}^{3}}{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,e}{3\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{4}{d}^{4}}}-12\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}{d}^{2}}}+6\,{\frac{e\sqrt{ex+d}}{{c}^{2}}}+{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{4} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{3}{d}^{2} \left ( cdex+a{e}^{2} \right ) }}+3\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{2} \left ( cdex+a{e}^{2} \right ) }}-{\frac{e{d}^{2}}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-7\,{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{4}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+21\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-21\,{\frac{a{e}^{3}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+7\,{\frac{e{d}^{2}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(11/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

2/5*e*(e*x+d)^(5/2)/c^2/d^2-4/3/c^3/d^3*(e*x+d)^(3/2)*a*e^3+4/3*e/c^2/d*(e*x+d)^
(3/2)+6/c^4/d^4*a^2*e^5*(e*x+d)^(1/2)-12/c^3/d^2*a*e^3*(e*x+d)^(1/2)+6*e/c^2*(e*
x+d)^(1/2)+1/c^4/d^4*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^3*e^7-3/c^3/d^2*(e*x+d)^(1/
2)/(c*d*e*x+a*e^2)*a^2*e^5+3/c^2*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a*e^3-e/c*d^2*(e*
x+d)^(1/2)/(c*d*e*x+a*e^2)-7/c^4/d^4/((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^7+21/c^3/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^2*e^5-21/c^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+7*e/c*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))

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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)^(11/2)/(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 = 0.226425, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac{105 \,{\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \,{\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(11/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

[1/30*(105*(a*c^2*d^4*e^2 - 2*a^2*c*d^2*e^4 + a^3*e^6 + (c^3*d^5*e - 2*a*c^2*d^3
*e^3 + a^2*c*d*e^5)*x)*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*(6*c^3*d
^3*e^3*x^3 - 15*c^3*d^6 + 161*a*c^2*d^4*e^2 - 245*a^2*c*d^2*e^4 + 105*a^3*e^6 +
2*(16*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + 2*(58*c^3*d^5*e - 84*a*c^2*d^3*e^3 +
35*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^5*d^5*x + a*c^4*d^4*e), -1/15*(105*(a*c^2*d
^4*e^2 - 2*a^2*c*d^2*e^4 + a^3*e^6 + (c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)
*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d
))) - (6*c^3*d^3*e^3*x^3 - 15*c^3*d^6 + 161*a*c^2*d^4*e^2 - 245*a^2*c*d^2*e^4 +
105*a^3*e^6 + 2*(16*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + 2*(58*c^3*d^5*e - 84*a*
c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^5*d^5*x + a*c^4*d^4*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)**(11/2)/(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)^(11/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Timed out