3.2000 \(\int \frac{(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=144 \[ -\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac{5 e \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2} \]

[Out]

(5*e*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(c^3*d^3) + (5*e*(d + e*x)^(3/2))/(3*c^2*d^2
) - (d + e*x)^(5/2)/(c*d*(a*e + c*d*x)) - (5*e*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sq
rt[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 = 0.260892, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac{5 e \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac{(d+e x)^{5/2}}{c d (a e+c d x)}+\frac{5 e (d+e x)^{3/2}}{3 c^2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(5*e*(c*d^2 - a*e^2)*Sqrt[d + e*x])/(c^3*d^3) + (5*e*(d + e*x)^(3/2))/(3*c^2*d^2
) - (d + e*x)^(5/2)/(c*d*(a*e + c*d*x)) - (5*e*(c*d^2 - a*e^2)^(3/2)*ArcTanh[(Sq
rt[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 = 60.8028, size = 129, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{\frac{5}{2}}}{c d \left (a e + c d x\right )} + \frac{5 e \left (d + e x\right )^{\frac{3}{2}}}{3 c^{2} d^{2}} - \frac{5 e \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )}{c^{3} d^{3}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-(d + e*x)**(5/2)/(c*d*(a*e + c*d*x)) + 5*e*(d + e*x)**(3/2)/(3*c**2*d**2) - 5*e
*sqrt(d + e*x)*(a*e**2 - c*d**2)/(c**3*d**3) + 5*e*(a*e**2 - c*d**2)**(3/2)*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 = 0.220094, size = 146, normalized size = 1.01 \[ -\frac{\sqrt{d+e x} \left (15 a^2 e^4+10 a c d e^2 (e x-2 d)+c^2 d^2 \left (3 d^2-14 d e x-2 e^2 x^2\right )\right )}{3 c^3 d^3 (a e+c d x)}-\frac{5 e \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(3*d^2 - 14*d*
e*x - 2*e^2*x^2)))/(3*c^3*d^3*(a*e + c*d*x)) - (5*e*(c*d^2 - a*e^2)^(3/2)*ArcTan
h[(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 = 0.021, size = 314, normalized size = 2.2 \[{\frac{2\,e}{3\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}{d}^{3}}}+4\,{\frac{e\sqrt{ex+d}}{{c}^{2}d}}-{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{3} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+2\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{2}d \left ( cdex+a{e}^{2} \right ) }}-{\frac{de}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+5\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}{d}^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-10\,{\frac{a{e}^{3}}{{c}^{2}d\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+5\,{\frac{de}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

2/3*e*(e*x+d)^(3/2)/c^2/d^2-4/c^3/d^3*a*e^3*(e*x+d)^(1/2)+4*e/c^2/d*(e*x+d)^(1/2
)-1/c^3/d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^2*e^5+2/c^2/d*(e*x+d)^(1/2)/(c*d*e*x
+a*e^2)*a*e^3-e/c*d*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+5/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^2*e^5-10/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*e^3+5
*e/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))

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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)^(9/2)/(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 = 0.232794, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \,{\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}, -\frac{15 \,{\left (a c d^{2} e^{2} - a^{2} e^{4} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \,{\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

[1/6*(15*(a*c*d^2*e^2 - a^2*e^4 + (c^2*d^3*e - a*c*d*e^3)*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*(2*c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 20*a*c*d^2*e^2 -
15*a^2*e^4 + 2*(7*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^4*d^4*x + a*c^3*
d^3*e), -1/3*(15*(a*c*d^2*e^2 - a^2*e^4 + (c^2*d^3*e - a*c*d*e^3)*x)*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^2*d^
2*e^2*x^2 - 3*c^2*d^4 + 20*a*c*d^2*e^2 - 15*a^2*e^4 + 2*(7*c^2*d^3*e - 5*a*c*d*e
^3)*x)*sqrt(e*x + d))/(c^4*d^4*x + a*c^3*d^3*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)**(9/2)/(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)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Timed out