3.363 \(\int \frac{(d+e x)^{3/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c} \]

[Out]

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

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Rubi [A]  time = 0.278711, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.2204, size = 80, normalized size = 0.87 \[ \frac{2 e \sqrt{d + e x}}{c} - \frac{2 d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b} - \frac{2 \left (b e - c d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.303355, size = 92, normalized size = 1. \[ \frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 159, normalized size = 1.7 \[ 2\,{\frac{e\sqrt{ex+d}}{c}}-2\,{\frac{b{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{de}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{3/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276623, size = 1, normalized size = 0.01 \[ \left [\frac{c d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} b e -{\left (c d - b e\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, \frac{c d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} b e + 2 \,{\left (c d - b e\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right )}{b c}, -\frac{2 \, c \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - 2 \, \sqrt{e x + d} b e +{\left (c d - b e\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, -\frac{2 \,{\left (c \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-d}}\right ) - \sqrt{e x + d} b e -{\left (c d - b e\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d - b e}{c}}}\right )\right )}}{b c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 26.1212, size = 265, normalized size = 2.88 \[ \frac{2 e \sqrt{d + e x}}{c} - \frac{2 d^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{\sqrt{- d}} & \text{for}\: - d > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: - d < 0 \wedge d < d + e x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{\sqrt{d}} & \text{for}\: d > d + e x \wedge - d < 0 \end{cases}\right )}{b} - \frac{2 \left (b e - c d\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{c \sqrt{\frac{b e - c d}{c}}} & \text{for}\: \frac{b e - c d}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: d + e x > \frac{- b e + c d}{c} \wedge \frac{b e - c d}{c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- b e + c d}{c}}} \right )}}{c \sqrt{\frac{- b e + c d}{c}}} & \text{for}\: \frac{b e - c d}{c} < 0 \wedge d + e x < \frac{- b e + c d}{c} \end{cases}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*sqrt(d + e*x)/c - 2*d**2*Piecewise((-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)), (atan
h(sqrt(d + e*x)/sqrt(d))/sqrt(d), (-d < 0) & (d > d + e*x)))/b - 2*(b*e - c*d)**
2*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)

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GIAC/XCAS [A]  time = 0.210732, size = 151, normalized size = 1.64 \[ \frac{2 \, d^{2} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \, \sqrt{x e + d} e}{c} - \frac{2 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*d^2*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*sqrt(x*e + d)*e/c - 2*(c^2
*d^2 - 2*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-
c^2*d + b*c*e)*b*c)