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

Optimal. Leaf size=251 \[ -\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac{e \sqrt{d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac{(d+e x)^{5/2} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{7/2}}{b x (b+c x)} \]

[Out]

(e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^3) + (e*(
6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*(d + e*x)^(3/2))/(3*b^2*c^2) - ((c*d - b*e)*(
2*c*d - b*e)*(d + e*x)^(5/2))/(b^2*c*(b + c*x)) - (d*(d + e*x)^(7/2))/(b*x*(b +
c*x)) + (d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b
*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3
*c^(7/2))

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Rubi [A]  time = 1.14487, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{e (d+e x)^{3/2} \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 c^2}+\frac{e \sqrt{d+e x} (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 c^3}-\frac{(d+e x)^{5/2} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{7/2}}{b x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)*Sqrt[d + e*x])/(b^2*c^3) + (e*(
6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2)*(d + e*x)^(3/2))/(3*b^2*c^2) - ((c*d - b*e)*(
2*c*d - b*e)*(d + e*x)^(5/2))/(b^2*c*(b + c*x)) - (d*(d + e*x)^(7/2))/(b*x*(b +
c*x)) + (d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b
*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3
*c^(7/2))

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Rubi in Sympy [A]  time = 120.03, size = 231, normalized size = 0.92 \[ - \frac{d \left (d + e x\right )^{\frac{7}{2}}}{b x \left (b + c x\right )} - \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{b^{2} c \left (b + c x\right )} + \frac{e \left (d + e x\right )^{\frac{3}{2}} \left (5 b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{3 b^{2} c^{2}} - \frac{e \sqrt{d + e x} \left (b e - 2 c d\right ) \left (5 b^{2} e^{2} - b c d e + c^{2} d^{2}\right )}{b^{2} c^{3}} - \frac{d^{\frac{7}{2}} \left (9 b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{\frac{7}{2}} \left (5 b e + 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*(d + e*x)**(7/2)/(b*x*(b + c*x)) - (d + e*x)**(5/2)*(b*e - 2*c*d)*(b*e - c*d)
/(b**2*c*(b + c*x)) + e*(d + e*x)**(3/2)*(5*b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)
/(3*b**2*c**2) - e*sqrt(d + e*x)*(b*e - 2*c*d)*(5*b**2*e**2 - b*c*d*e + c**2*d**
2)/(b**2*c**3) - d**(7/2)*(9*b*e - 4*c*d)*atanh(sqrt(d + e*x)/sqrt(d))/b**3 + (b
*e - c*d)**(7/2)*(5*b*e + 4*c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*
*3*c**(7/2))

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Mathematica [A]  time = 0.488, size = 171, normalized size = 0.68 \[ -\frac{(c d-b e)^{7/2} (5 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}+\frac{d^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\sqrt{d+e x} \left (-\frac{(c d-b e)^4}{b^2 c^3 (b+c x)}-\frac{d^4}{b^2 x}+\frac{2 e^3 (13 c d-6 b e)}{3 c^3}+\frac{2 e^4 x}{3 c^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((2*e^3*(13*c*d - 6*b*e))/(3*c^3) - d^4/(b^2*x) + (2*e^4*x)/(3*c^2
) - (c*d - b*e)^4/(b^2*c^3*(b + c*x))) + (d^(7/2)*(4*c*d - 9*b*e)*ArcTanh[Sqrt[d
+ e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*c*d + 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(7/2))

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Maple [B]  time = 0.039, size = 515, normalized size = 2.1 \[{\frac{2\,{e}^{3}}{3\,{c}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{e}^{4}b\sqrt{ex+d}}{{c}^{3}}}+8\,{\frac{{e}^{3}d\sqrt{ex+d}}{{c}^{2}}}-{\frac{{e}^{5}{b}^{2}}{{c}^{3} \left ( cex+be \right ) }\sqrt{ex+d}}+4\,{\frac{{e}^{4}b\sqrt{ex+d}d}{{c}^{2} \left ( cex+be \right ) }}-6\,{\frac{{e}^{3}\sqrt{ex+d}{d}^{2}}{c \left ( cex+be \right ) }}+4\,{\frac{{e}^{2}\sqrt{ex+d}{d}^{3}}{b \left ( cex+be \right ) }}-{\frac{ce{d}^{4}}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{{e}^{5}{b}^{2}}{{c}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-16\,{\frac{{e}^{4}bd}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+14\,{\frac{{e}^{3}{d}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{e}^{2}{d}^{3}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-11\,{\frac{ce{d}^{4}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{5}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{4}}{{b}^{2}x}\sqrt{ex+d}}-9\,{\frac{e{d}^{7/2}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{9/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*e^3/c^2*(e*x+d)^(3/2)-4*e^4/c^3*b*(e*x+d)^(1/2)+8*e^3/c^2*d*(e*x+d)^(1/2)-e^
5/c^3*b^2*(e*x+d)^(1/2)/(c*e*x+b*e)+4*e^4/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)*d-6*e^
3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*d^2+4*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*d^3-e*c/b^2*
(e*x+d)^(1/2)/(c*e*x+b*e)*d^4+5*e^5/c^3*b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2))-16*e^4/c^2*b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1
/2)/((b*e-c*d)*c)^(1/2))*d+14*e^3/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/(
(b*e-c*d)*c)^(1/2))*d^2+4*e^2/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e
-c*d)*c)^(1/2))*d^3-11*e*c/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*d^4+4*c^2/b^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*
d)*c)^(1/2))*d^5-d^4/b^2*(e*x+d)^(1/2)/x-9*e*d^(7/2)/b^2*arctanh((e*x+d)^(1/2)/d
^(1/2))+4*d^(9/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*c

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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*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.08674, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*
b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d
*e^3 - 5*b^5*e^4)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x +
d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*
b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d
)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)
*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 1
5*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), -1/6*(6*((4*c^5*d^4 - 7*
b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4
*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt
(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + 3*((4*c^5*d^4 - 9*
b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(d)*log((e*x - 2*sqrt(
e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^
2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^
2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/
6*(6*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 + (4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(
-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - 3*((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d
^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e -
3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*b^5*e^4)*x)*sqrt((c*d - b*e)/c)*log((c*e*
x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(2*b^3*c
^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b*c^4*d
^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*x)*sqr
t(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x), 1/3*(3*((4*c^5*d^4 - 9*b*c^4*d^3*e)*x^2 +
(4*b*c^4*d^4 - 9*b^2*c^3*d^3*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d)) - 3*
((4*c^5*d^4 - 7*b*c^4*d^3*e - 3*b^2*c^3*d^2*e^2 + 11*b^3*c^2*d*e^3 - 5*b^4*c*e^4
)*x^2 + (4*b*c^4*d^4 - 7*b^2*c^3*d^3*e - 3*b^3*c^2*d^2*e^2 + 11*b^4*c*d*e^3 - 5*
b^5*e^4)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (2
*b^3*c^2*e^4*x^3 - 3*b^2*c^3*d^4 + 2*(13*b^3*c^2*d*e^3 - 5*b^4*c*e^4)*x^2 - (6*b
*c^4*d^4 - 12*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 38*b^4*c*d*e^3 + 15*b^5*e^4)*
x)*sqrt(e*x + d))/(b^3*c^4*x^2 + b^4*c^3*x)]

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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)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24076, size = 589, normalized size = 2.35 \[ -\frac{{\left (4 \, c d^{5} - 9 \, b d^{4} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (4 \, c^{5} d^{5} - 11 \, b c^{4} d^{4} e + 4 \, b^{2} c^{3} d^{3} e^{2} + 14 \, b^{3} c^{2} d^{2} e^{3} - 16 \, b^{4} c d e^{4} + 5 \, b^{5} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{4} e^{3} + 12 \, \sqrt{x e + d} c^{4} d e^{3} - 6 \, \sqrt{x e + d} b c^{3} e^{4}\right )}}{3 \, c^{6}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{4} d^{4} e - 2 \, \sqrt{x e + d} c^{4} d^{5} e - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{3} d^{3} e^{2} + 5 \, \sqrt{x e + d} b c^{3} d^{4} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{2} e^{3} - 6 \, \sqrt{x e + d} b^{2} c^{2} d^{3} e^{3} - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c d e^{4} + 4 \, \sqrt{x e + d} b^{3} c d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{5} - \sqrt{x e + d} b^{4} d e^{5}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*c*d^5 - 9*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^5*d^
5 - 11*b*c^4*d^4*e + 4*b^2*c^3*d^3*e^2 + 14*b^3*c^2*d^2*e^3 - 16*b^4*c*d*e^4 + 5
*b^5*e^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3
*c^3) + 2/3*((x*e + d)^(3/2)*c^4*e^3 + 12*sqrt(x*e + d)*c^4*d*e^3 - 6*sqrt(x*e +
d)*b*c^3*e^4)/c^6 - (2*(x*e + d)^(3/2)*c^4*d^4*e - 2*sqrt(x*e + d)*c^4*d^5*e -
4*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 5*sqrt(x*e + d)*b*c^3*d^4*e^2 + 6*(x*e + d)^(3
/2)*b^2*c^2*d^2*e^3 - 6*sqrt(x*e + d)*b^2*c^2*d^3*e^3 - 4*(x*e + d)^(3/2)*b^3*c*
d*e^4 + 4*sqrt(x*e + d)*b^3*c*d^2*e^4 + (x*e + d)^(3/2)*b^4*e^5 - sqrt(x*e + d)*
b^4*d*e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^
2*c^3)