3.362 \(\int \frac{(d+e x)^{5/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=118 \[ \frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.477672, antiderivative size = 118, 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 \[ \frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}+\frac{2 e \sqrt{d+e x} (2 c d-b e)}{c^2}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e (d+e x)^{3/2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.7811, size = 105, normalized size = 0.89 \[ \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} - \frac{2 e \sqrt{d + e x} \left (b e - 2 c d\right )}{c^{2}} - \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} + \frac{2 \left (b e - c d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.27291, size = 107, normalized size = 0.91 \[ \frac{2 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}+\frac{2 e \sqrt{d+e x} (-3 b e+7 c d+c e x)}{3 c^2}-\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 237, normalized size = 2. \[{\frac{2\,e}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{b{e}^{2}\sqrt{ex+d}}{{c}^{2}}}+4\,{\frac{de\sqrt{ex+d}}{c}}+2\,{\frac{{b}^{2}{e}^{3}}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-6\,{\frac{bd{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+6\,{\frac{e{d}^{2}}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{3}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{5/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*e*(e*x+d)^(3/2)/c-2/c^2*b*e^2*(e*x+d)^(1/2)+4*d*e*(e*x+d)^(1/2)/c+2/c^2*b^2*
e^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-6/c*b*e^2/((
b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d+6*e/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^2-2*c/b/((b*e-c*d)*c)^(1/2
)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^3-2*d^(5/2)*arctanh((e*x+d)^(1/2
)/d^(1/2))/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.390083, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} d^{\frac{5}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) + 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{3 \, c^{2} d^{\frac{5}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 6 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right ) + 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, -\frac{6 \, c^{2} \sqrt{-d} d^{2} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right ) - 2 \,{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}}{3 \, b c^{2}}, -\frac{2 \,{\left (3 \, c^{2} \sqrt{-d} d^{2} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - 3 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right ) -{\left (b c e^{2} x + 7 \, b c d e - 3 \, b^{2} e^{2}\right )} \sqrt{e x + d}\right )}}{3 \, b c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 47.2429, size = 294, normalized size = 2.49 \[ \frac{2 e \left (d + e x\right )^{\frac{3}{2}}}{3 c} + \frac{\sqrt{d + e x} \left (- 2 b e^{2} + 4 c d e\right )}{c^{2}} - \frac{2 d^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} & \text{for}\: - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: - d < 0 \wedge d < d + e x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: d > d + e x \wedge - d < 0 \end{cases}\right )}{b} + \frac{2 \left (b e - c d\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{c \sqrt{\frac{b e - c d}{c}}} & \text{for}\: \frac{b e - c d}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: d + e x > \frac{- b e + c d}{c} \wedge \frac{b e - c d}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: \frac{b e - c d}{c} < 0 \wedge d + e x < \frac{- b e + c d}{c} \end{cases}\right )}{b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(d + e*x)**(3/2)/(3*c) + sqrt(d + e*x)*(-2*b*e**2 + 4*c*d*e)/c**2 - 2*d**3*P
iecewise((-atan(sqrt(d + e*x)/sqrt(-d))/sqrt(-d), -d > 0), (acoth(sqrt(d + e*x)/
sqrt(d))/sqrt(d), (-d < 0) & (d < d + e*x)), (atanh(sqrt(d + e*x)/sqrt(d))/sqrt(
d), (-d < 0) & (d > d + e*x)))/b + 2*(b*e - c*d)**3*Piecewise((atan(sqrt(d + e*x
)/sqrt((b*e - c*d)/c))/(c*sqrt((b*e - c*d)/c)), (b*e - c*d)/c > 0), (-acoth(sqrt
(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sqrt((-b*e + c*d)/c)), ((b*e - c*d)/c < 0) &
(d + e*x > (-b*e + c*d)/c)), (-atanh(sqrt(d + e*x)/sqrt((-b*e + c*d)/c))/(c*sqrt
((-b*e + c*d)/c)), ((b*e - c*d)/c < 0) & (d + e*x < (-b*e + c*d)/c)))/(b*c**2)

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GIAC/XCAS [A]  time = 0.2134, size = 217, normalized size = 1.84 \[ \frac{2 \, d^{3} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} - \frac{2 \,{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e + 6 \, \sqrt{x e + d} c^{2} d e - 3 \, \sqrt{x e + d} b c e^{2}\right )}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d^3*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) - 2*(c^3*d^3 - 3*b*c^2*d^2*e +
3*b^2*c*d*e^2 - b^3*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^
2*d + b*c*e)*b*c^2) + 2/3*((x*e + d)^(3/2)*c^2*e + 6*sqrt(x*e + d)*c^2*d*e - 3*s
qrt(x*e + d)*b*c*e^2)/c^3