3.375 \(\int \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=216 \[ -\frac{c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}+\frac{(3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{5/2}}-\frac{e \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{c (2 c d-b e)}{b^2 d (b+c x) \sqrt{d+e x} (c d-b e)}-\frac{1}{b d x (b+c x) \sqrt{d+e x}} \]

[Out]

-((e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2))/(b^2*d^2*(c*d - b*e)^2*Sqrt[d + e*x]))
- (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*Sqrt[d + e*x]) - 1/(b*d*x*(b +
c*x)*Sqrt[d + e*x]) + ((4*c*d + 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(
5/2)) - (c^(5/2)*(4*c*d - 7*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]
])/(b^3*(c*d - b*e)^(5/2))

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Rubi [A]  time = 0.910895, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}+\frac{(3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{5/2}}-\frac{e \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{c (2 c d-b e)}{b^2 d (b+c x) \sqrt{d+e x} (c d-b e)}-\frac{1}{b d x (b+c x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^2),x]

[Out]

-((e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2))/(b^2*d^2*(c*d - b*e)^2*Sqrt[d + e*x]))
- (c*(2*c*d - b*e))/(b^2*d*(c*d - b*e)*(b + c*x)*Sqrt[d + e*x]) - 1/(b*d*x*(b +
c*x)*Sqrt[d + e*x]) + ((4*c*d + 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(
5/2)) - (c^(5/2)*(4*c*d - 7*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]
])/(b^3*(c*d - b*e)^(5/2))

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Rubi in Sympy [A]  time = 108.085, size = 194, normalized size = 0.9 \[ - \frac{c}{b x \left (b + c x\right ) \sqrt{d + e x} \left (b e - c d\right )} - \frac{b e - 2 c d}{b^{2} d x \sqrt{d + e x} \left (b e - c d\right )} - \frac{e \left (3 b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{b^{2} d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} - \frac{c^{\frac{5}{2}} \left (7 b e - 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \left (b e - c d\right )^{\frac{5}{2}}} + \frac{\left (3 b e + 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**2,x)

[Out]

-c/(b*x*(b + c*x)*sqrt(d + e*x)*(b*e - c*d)) - (b*e - 2*c*d)/(b**2*d*x*sqrt(d +
e*x)*(b*e - c*d)) - e*(3*b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(b**2*d**2*sqrt(d
+ e*x)*(b*e - c*d)**2) - c**(5/2)*(7*b*e - 4*c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqr
t(b*e - c*d))/(b**3*(b*e - c*d)**(5/2)) + (3*b*e + 4*c*d)*atanh(sqrt(d + e*x)/sq
rt(d))/(b**3*d**(5/2))

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Mathematica [A]  time = 0.931046, size = 164, normalized size = 0.76 \[ -\frac{c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}+\frac{(3 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{5/2}}+\sqrt{d+e x} \left (-\frac{\frac{c^3}{(b+c x) (c d-b e)^2}+\frac{1}{d^2 x}}{b^2}-\frac{2 e^3}{d^2 (d+e x) (c d-b e)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^2),x]

[Out]

Sqrt[d + e*x]*((-2*e^3)/(d^2*(c*d - b*e)^2*(d + e*x)) - (1/(d^2*x) + c^3/((c*d -
b*e)^2*(b + c*x)))/b^2) + ((4*c*d + 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3
*d^(5/2)) - (c^(5/2)*(4*c*d - 7*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(b^3*(c*d - b*e)^(5/2))

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Maple [A]  time = 0.035, size = 229, normalized size = 1.1 \[ -2\,{\frac{{e}^{3}}{{d}^{2} \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}-{\frac{e{c}^{3}}{{b}^{2} \left ( be-cd \right ) ^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-7\,{\frac{e{c}^{3}}{{b}^{2} \left ( be-cd \right ) ^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{4}d}{{b}^{3} \left ( be-cd \right ) ^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}{d}^{2}x}\sqrt{ex+d}}+3\,{\frac{e}{{b}^{2}{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{c}{{b}^{3}{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^2,x)

[Out]

-2*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(1/2)-e*c^3/b^2/(b*e-c*d)^2*(e*x+d)^(1/2)/(c*e*x+
b*e)-7*e*c^3/b^2/(b*e-c*d)^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*
d)*c)^(1/2))+4*c^4/b^3/(b*e-c*d)^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((
b*e-c*d)*c)^(1/2))*d-1/b^2/d^2*(e*x+d)^(1/2)/x+3*e/b^2/d^(5/2)*arctanh((e*x+d)^(
1/2)/d^(1/2))+4/b^3/d^(3/2)*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(1/((c*x^2 + b*x)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.777609, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/2*(((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqr
t(e*x + d)*sqrt(d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*
sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - ((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*
b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^
2 + 3*b^4*e^3)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x)
+ 2*(b^2*c^2*d^3 - 2*b^3*c*d^2*e + b^4*d*e^2 + (2*b*c^3*d^2*e - 2*b^2*c^2*d*e^2
+ 3*b^3*c*e^3)*x^2 + (2*b*c^3*d^3 - b^2*c^2*d^2*e - b^3*c*d*e^2 + 3*b^4*e^3)*x)
*sqrt(d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 -
2*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(d)), -1/2*(2*((4*c^4*d^3 - 7
*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(e*x + d)*sqrt(d)*sqr
t(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) -
((4*c^4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3
- 5*b^2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(e*x + d)*log(((e*x + 2*d)
*sqrt(d) + 2*sqrt(e*x + d)*d)/x) + 2*(b^2*c^2*d^3 - 2*b^3*c*d^2*e + b^4*d*e^2 +
(2*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (2*b*c^3*d^3 - b^2*c^2*d^2
*e - b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5
*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*s
qrt(d)), -1/2*(((4*c^4*d^3 - 7*b*c^3*d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e
)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*
d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*((4*c^4*d^3 - 5*b*c^3
*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^2*c^2*d^2*e - 2
*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) +
2*(b^2*c^2*d^3 - 2*b^3*c*d^2*e + b^4*d*e^2 + (2*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 +
3*b^3*c*e^3)*x^2 + (2*b*c^3*d^3 - b^2*c^2*d^2*e - b^3*c*d*e^2 + 3*b^4*e^3)*x)*sq
rt(-d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 - 2
*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(-d)), -(((4*c^4*d^3 - 7*b*c^3*
d^2*e)*x^2 + (4*b*c^3*d^3 - 7*b^2*c^2*d^2*e)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(
c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + ((4*c^
4*d^3 - 5*b*c^3*d^2*e - 2*b^2*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (4*b*c^3*d^3 - 5*b^
2*c^2*d^2*e - 2*b^3*c*d*e^2 + 3*b^4*e^3)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d
)*sqrt(-d))) + (b^2*c^2*d^3 - 2*b^3*c*d^2*e + b^4*d*e^2 + (2*b*c^3*d^2*e - 2*b^2
*c^2*d*e^2 + 3*b^3*c*e^3)*x^2 + (2*b*c^3*d^3 - b^2*c^2*d^2*e - b^3*c*d*e^2 + 3*b
^4*e^3)*x)*sqrt(-d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^
4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**2,x)

[Out]

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.215967, size = 477, normalized size = 2.21 \[ \frac{{\left (4 \, c^{4} d - 7 \, b c^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{2} c^{3} d^{2} e - 2 \,{\left (x e + d\right )} c^{3} d^{3} e - 2 \,{\left (x e + d\right )}^{2} b c^{2} d e^{2} + 3 \,{\left (x e + d\right )} b c^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{2} b^{2} c e^{3} - 7 \,{\left (x e + d\right )} b^{2} c d e^{3} + 2 \, b^{2} c d^{2} e^{3} + 3 \,{\left (x e + d\right )} b^{3} e^{4} - 2 \, b^{3} d e^{4}}{{\left (b^{2} c^{2} d^{4} - 2 \, b^{3} c d^{3} e + b^{4} d^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{5}{2}} c - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} c d + \sqrt{x e + d} c d^{2} +{\left (x e + d\right )}^{\frac{3}{2}} b e - \sqrt{x e + d} b d e\right )}} - \frac{{\left (4 \, c d + 3 \, b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

(4*c^4*d - 7*b*c^3*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^2*d^2
- 2*b^4*c*d*e + b^5*e^2)*sqrt(-c^2*d + b*c*e)) - (2*(x*e + d)^2*c^3*d^2*e - 2*(
x*e + d)*c^3*d^3*e - 2*(x*e + d)^2*b*c^2*d*e^2 + 3*(x*e + d)*b*c^2*d^2*e^2 + 3*(
x*e + d)^2*b^2*c*e^3 - 7*(x*e + d)*b^2*c*d*e^3 + 2*b^2*c*d^2*e^3 + 3*(x*e + d)*b
^3*e^4 - 2*b^3*d*e^4)/((b^2*c^2*d^4 - 2*b^3*c*d^3*e + b^4*d^2*e^2)*((x*e + d)^(5
/2)*c - 2*(x*e + d)^(3/2)*c*d + sqrt(x*e + d)*c*d^2 + (x*e + d)^(3/2)*b*e - sqrt
(x*e + d)*b*d*e)) - (4*c*d + 3*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)
*d^2)