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

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Rubi [A]  time = 0.195558, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.19, Rules used = {703, 826, 1166, 208} $\frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c}$

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

Rule 703

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*x^2),
x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*
e, 0] && GtQ[m, 1]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{3/2}}{b x+c x^2} \, dx &=\frac{2 e \sqrt{d+e x}}{c}+\frac{\int \frac{c d^2+e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{c}\\ &=\frac{2 e \sqrt{d+e x}}{c}+\frac{2 \operatorname{Subst}\left (\int \frac{c d^2 e-d e (2 c d-b e)+e (2 c d-b e) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c}\\ &=\frac{2 e \sqrt{d+e x}}{c}+\frac{\left (2 c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}-\frac{\left (2 (c d-b e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b c}\\ &=\frac{2 e \sqrt{d+e x}}{c}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\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}}\\ \end{align*}

Mathematica [A]  time = 0.0605512, size = 97, normalized size = 1.05 $\frac{2 \left (b \sqrt{c} e \sqrt{d+e x}+(c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )-c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{b c^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.223, size = 159, normalized size = 1.7 \begin{align*} 2\,{\frac{e\sqrt{ex+d}}{c}}-2\,{\frac{b{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{de}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{3/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

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((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))+4*e/((b*e-c*d)*c
)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d-2/b*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 2.60947, size = 1007, normalized size = 10.95 \begin{align*} \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} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right )}{b c}, \frac{2 \, c \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{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}}{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} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right )\right )}}{b c}\right ] \end{align*}

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

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Sympy [A]  time = 33.7844, size = 92, normalized size = 1. \begin{align*} \frac{2 e \sqrt{d + e x}}{c} + \frac{2 d^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} - \frac{2 \left (b e - c d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b c^{2} \sqrt{\frac{b e - c d}{c}}} \end{align*}

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*atan(sqrt(d + e*x)/sqrt(-d))/(b*sqrt(-d)) - 2*(b*e - c*d)**2*atan(sqrt(d + e*x)/s
qrt((b*e - c*d)/c))/(b*c**2*sqrt((b*e - c*d)/c))

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Giac [A]  time = 1.37459, size = 151, normalized size = 1.64 \begin{align*} \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} \end{align*}

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