3.1998 \(\int \frac{(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2} \]

[Out]

(9*e*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/(c^5*d^5) + (3*e*(c*d^2 - a*e^2)^2*(d + e*
x)^(3/2))/(c^4*d^4) + (9*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2))/(5*c^3*d^3) + (9*e*(
d + e*x)^(7/2))/(7*c^2*d^2) - (d + e*x)^(9/2)/(c*d*(a*e + c*d*x)) - (9*e*(c*d^2
- a*e^2)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^
(11/2)*d^(11/2))

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Rubi [A]  time = 0.436822, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(9*e*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/(c^5*d^5) + (3*e*(c*d^2 - a*e^2)^2*(d + e*
x)^(3/2))/(c^4*d^4) + (9*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2))/(5*c^3*d^3) + (9*e*(
d + e*x)^(7/2))/(7*c^2*d^2) - (d + e*x)^(9/2)/(c*d*(a*e + c*d*x)) - (9*e*(c*d^2
- a*e^2)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^
(11/2)*d^(11/2))

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Rubi in Sympy [A]  time = 95.8492, size = 192, normalized size = 0.91 \[ - \frac{\left (d + e x\right )^{\frac{9}{2}}}{c d \left (a e + c d x\right )} + \frac{9 e \left (d + e x\right )^{\frac{7}{2}}}{7 c^{2} d^{2}} - \frac{9 e \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 c^{3} d^{3}} + \frac{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}}{c^{4} d^{4}} - \frac{9 e \sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3}}{c^{5} d^{5}} + \frac{9 e \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{11}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-(d + e*x)**(9/2)/(c*d*(a*e + c*d*x)) + 9*e*(d + e*x)**(7/2)/(7*c**2*d**2) - 9*e
*(d + e*x)**(5/2)*(a*e**2 - c*d**2)/(5*c**3*d**3) + 3*e*(d + e*x)**(3/2)*(a*e**2
- c*d**2)**2/(c**4*d**4) - 9*e*sqrt(d + e*x)*(a*e**2 - c*d**2)**3/(c**5*d**5) +
9*e*(a*e**2 - c*d**2)**(7/2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c
*d**2))/(c**(11/2)*d**(11/2))

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Mathematica [A]  time = 0.417846, size = 246, normalized size = 1.17 \[ -\frac{\sqrt{d+e x} \left (315 a^4 e^8-210 a^3 c d e^6 (5 d-e x)+42 a^2 c^2 d^2 e^4 \left (29 d^2-17 d e x-e^2 x^2\right )+6 a c^3 d^3 e^2 \left (-88 d^3+142 d^2 e x+23 d e^2 x^2+3 e^3 x^3\right )+c^4 d^4 \left (35 d^4-388 d^3 e x-156 d^2 e^2 x^2-58 d e^3 x^3-10 e^4 x^4\right )\right )}{35 c^5 d^5 (a e+c d x)}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]*(315*a^4*e^8 - 210*a^3*c*d*e^6*(5*d - e*x) + 42*a^2*c^2*d^2*e^4*
(29*d^2 - 17*d*e*x - e^2*x^2) + 6*a*c^3*d^3*e^2*(-88*d^3 + 142*d^2*e*x + 23*d*e^
2*x^2 + 3*e^3*x^3) + c^4*d^4*(35*d^4 - 388*d^3*e*x - 156*d^2*e^2*x^2 - 58*d*e^3*
x^3 - 10*e^4*x^4)))/(35*c^5*d^5*(a*e + c*d*x)) - (9*e*(c*d^2 - a*e^2)^(7/2)*ArcT
anh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(11/2)*d^(11/2))

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Maple [B]  time = 0.024, size = 628, normalized size = 3. \[{\frac{2\,e}{7\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{3}}{5\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,e}{5\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{5}}{{c}^{4}{d}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}a{e}^{3}}{{c}^{3}{d}^{2}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}}{{c}^{2}}}-8\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{5}{d}^{5}}}+24\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{4}{d}^{3}}}-24\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}d}}+8\,{\frac{de\sqrt{ex+d}}{{c}^{2}}}-{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+4\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{4}{d}^{3} \left ( cdex+a{e}^{2} \right ) }}-6\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{3}d \left ( cdex+a{e}^{2} \right ) }}+4\,{\frac{d\sqrt{ex+d}a{e}^{3}}{{c}^{2} \left ( cdex+a{e}^{2} \right ) }}-{\frac{e{d}^{3}}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-36\,{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+54\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}d\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-36\,{\frac{ad{e}^{3}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+9\,{\frac{e{d}^{3}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(13/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

2/7*e*(e*x+d)^(7/2)/c^2/d^2-4/5/c^3/d^3*(e*x+d)^(5/2)*a*e^3+4/5*e/c^2/d*(e*x+d)^
(5/2)+2/c^4/d^4*(e*x+d)^(3/2)*a^2*e^5-4/c^3/d^2*(e*x+d)^(3/2)*a*e^3+2*e/c^2*(e*x
+d)^(3/2)-8/c^5/d^5*a^3*e^7*(e*x+d)^(1/2)+24/c^4/d^3*a^2*e^5*(e*x+d)^(1/2)-24/c^
3/d*a*e^3*(e*x+d)^(1/2)+8*e/c^2*d*(e*x+d)^(1/2)-1/c^5/d^5*(e*x+d)^(1/2)/(c*d*e*x
+a*e^2)*a^4*e^9+4/c^4/d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^3*e^7-6/c^3/d*(e*x+d)^
(1/2)/(c*d*e*x+a*e^2)*a^2*e^5+4/c^2*d*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a*e^3-e/c*d^
3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+9/c^5/d^5/((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^4*e^9-36/c^4/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^7+54/c^3/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^5-
36/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+9*e/c*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)^(13/2)/(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 = 0.229524, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) -{\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(13/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

[1/70*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4
*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*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*(10*c^4*d^4*e^4*x^4 - 35*c^4*d^8 + 528*a*c^3*d^6
*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e
^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2
*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5 - 105*a^3
*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*e), -1/35*(315*(a*c^3*d^6*e^2
- 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3
+ 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e
*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d))) - (10*c^4*d^4*e^4*x^4 - 35*c^4*d^8 + 528*a
*c^3*d^6*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c
^4*d^5*e^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2
*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5 -
105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*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)**(13/2)/(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)^(13/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Timed out