3.367 \(\int \frac{1}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=138 \[ \frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}-\frac{2 e (2 c d-b e)}{d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

[Out]

(-2*e)/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e))/(d^2*(c*d - b*e)^
2*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2)) + (2*c^(5/2)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(5/2))

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Rubi [A]  time = 0.539019, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}-\frac{2 e (2 c d-b e)}{d^2 \sqrt{d+e x} (c d-b e)^2}-\frac{2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)),x]

[Out]

(-2*e)/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e))/(d^2*(c*d - b*e)^
2*Sqrt[d + e*x]) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2)) + (2*c^(5/2)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(c*d - b*e)^(5/2))

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Rubi in Sympy [A]  time = 64.6832, size = 121, normalized size = 0.88 \[ \frac{2 e}{3 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} + \frac{2 e \left (b e - 2 c d\right )}{d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} - \frac{2 c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \left (b e - c d\right )^{\frac{5}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.859422, size = 128, normalized size = 0.93 \[ \frac{2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{5/2}}+\frac{2 e (b e (4 d+3 e x)-c d (7 d+6 e x))}{3 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.026, size = 147, normalized size = 1.1 \[ 2\,{\frac{b{e}^{2}}{{d}^{2} \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}-4\,{\frac{ce}{d \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}+{\frac{2\,e}{3\,d \left ( be-cd \right ) } \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{{c}^{3}}{ \left ( be-cd \right ) ^{2}b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(c*x^2+b*x),x)

[Out]

2/d^2/(b*e-c*d)^2/(e*x+d)^(1/2)*b*e^2-4*e/d/(b*e-c*d)^2/(e*x+d)^(1/2)*c+2/3*e/d/
(b*e-c*d)/(e*x+d)^(3/2)-2/(b*e-c*d)^2*c^3/b/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2))-2*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(5/2)

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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(1/((c*x^2 + b*x)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.38204, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (b + c x\right ) \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x),x)

[Out]

Integral(1/(x*(b + c*x)*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.211974, size = 235, normalized size = 1.7 \[ -\frac{2 \, c^{3} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} c d e + c d^{2} e - 3 \,{\left (x e + d\right )} b e^{2} - b d e^{2}\right )}}{3 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} + \frac{2 \, \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-2*c^3*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^2*d^2 - 2*b^2*c*d*e +
b^3*e^2)*sqrt(-c^2*d + b*c*e)) - 2/3*(6*(x*e + d)*c*d*e + c*d^2*e - 3*(x*e + d)*
b*e^2 - b*d*e^2)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*(x*e + d)^(3/2)) + 2*arc
tan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)*d^2)