3.1231 \(\int \frac{(A+B x) (d+e x)^{3/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=131 \[ \frac{2 \sqrt{d+e x} (A c e-b B e+B c d)}{c^2}-\frac{2 (b B-A c) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 A d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{3/2}}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.295819, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {824, 826, 1166, 208} \[ \frac{2 \sqrt{d+e x} (A c e-b B e+B c d)}{c^2}-\frac{2 (b B-A c) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 A d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]

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{(A+B x) (d+e x)^{3/2}}{b x+c x^2} \, dx &=\frac{2 B (d+e x)^{3/2}}{3 c}+\frac{\int \frac{\sqrt{d+e x} (A c d+(B c d-b B e+A c e) x)}{b x+c x^2} \, dx}{c}\\ &=\frac{2 (B c d-b B e+A c e) \sqrt{d+e x}}{c^2}+\frac{2 B (d+e x)^{3/2}}{3 c}+\frac{\int \frac{A c^2 d^2+\left (B (c d-b e)^2+A c e (2 c d-b e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{c^2}\\ &=\frac{2 (B c d-b B e+A c e) \sqrt{d+e x}}{c^2}+\frac{2 B (d+e x)^{3/2}}{3 c}+\frac{2 \operatorname{Subst}\left (\int \frac{A c^2 d^2 e-d \left (B (c d-b e)^2+A c e (2 c d-b e)\right )+\left (B (c d-b e)^2+A c e (2 c d-b e)\right ) 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^2}\\ &=\frac{2 (B c d-b B e+A c e) \sqrt{d+e x}}{c^2}+\frac{2 B (d+e x)^{3/2}}{3 c}+\frac{\left (2 A 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 (b B-A c) (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^2}\\ &=\frac{2 (B c d-b B e+A c e) \sqrt{d+e x}}{c^2}+\frac{2 B (d+e x)^{3/2}}{3 c}-\frac{2 A d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}-\frac{2 (b B-A c) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.155343, size = 123, normalized size = 0.94 \[ \frac{2 \sqrt{d+e x} (3 A c e+B (-3 b e+4 c d+c e x))}{3 c^2}+\frac{2 (A c-b B) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{5/2}}-\frac{2 A d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.016, size = 335, normalized size = 2.6 \begin{align*}{\frac{2\,B}{3\,c} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{Ae\sqrt{ex+d}}{c}}-2\,{\frac{bBe\sqrt{ex+d}}{{c}^{2}}}+2\,{\frac{Bd\sqrt{ex+d}}{c}}-2\,{\frac{A{d}^{3/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{Ab{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{Ade}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{Ac{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{B{e}^{2}{b}^{2}}{{c}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{bBde}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+2\,{\frac{B{d}^{2}}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x),x)

[Out]

2/3*B*(e*x+d)^(3/2)/c+2/c*A*e*(e*x+d)^(1/2)-2/c^2*b*B*e*(e*x+d)^(1/2)+2/c*B*d*(e*x+d)^(1/2)-2*A*d^(3/2)*arctan
h((e*x+d)^(1/2)/d^(1/2))/b-2*b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*e^2+4/((b*e
-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d*e-2/b*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2
)*c/((b*e-c*d)*c)^(1/2))*A*d^2+2*b^2/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*e^2
-4*b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d*e+2/((b*e-c*d)*c)^(1/2)*arctan((e*x
+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2

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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((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.63672, size = 1453, normalized size = 11.09 \begin{align*} \left [\frac{3 \, A c^{2} d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) - 3 \,{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} 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 ) + 2 \,{\left (B b c e x + 4 \, B b c d - 3 \,{\left (B b^{2} - A b c\right )} e\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{3 \, A c^{2} d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) - 6 \,{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} 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 ) + 2 \,{\left (B b c e x + 4 \, B b c d - 3 \,{\left (B b^{2} - A b c\right )} e\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{6 \, A c^{2} \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) - 3 \,{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} 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 ) + 2 \,{\left (B b c e x + 4 \, B b c d - 3 \,{\left (B b^{2} - A b c\right )} e\right )} \sqrt{e x + d}}{3 \, b c^{2}}, \frac{2 \,{\left (3 \, A c^{2} \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) - 3 \,{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} 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 ) +{\left (B b c e x + 4 \, B b c d - 3 \,{\left (B b^{2} - A b c\right )} e\right )} \sqrt{e x + d}\right )}}{3 \, b c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

[1/3*(3*A*c^2*d^(3/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 3*((B*b*c - A*c^2)*d - (B*b^2 - A*b*c)*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)) + 2*(B*b*c*e
*x + 4*B*b*c*d - 3*(B*b^2 - A*b*c)*e)*sqrt(e*x + d))/(b*c^2), 1/3*(3*A*c^2*d^(3/2)*log((e*x - 2*sqrt(e*x + d)*
sqrt(d) + 2*d)/x) - 6*((B*b*c - A*c^2)*d - (B*b^2 - A*b*c)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqr
t(-(c*d - b*e)/c)/(c*d - b*e)) + 2*(B*b*c*e*x + 4*B*b*c*d - 3*(B*b^2 - A*b*c)*e)*sqrt(e*x + d))/(b*c^2), 1/3*(
6*A*c^2*sqrt(-d)*d*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((B*b*c - A*c^2)*d - (B*b^2 - A*b*c)*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)) + 2*(B*b*c*e*x + 4*B*b*c*d
- 3*(B*b^2 - A*b*c)*e)*sqrt(e*x + d))/(b*c^2), 2/3*(3*A*c^2*sqrt(-d)*d*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((
B*b*c - A*c^2)*d - (B*b^2 - A*b*c)*e)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d -
b*e)) + (B*b*c*e*x + 4*B*b*c*d - 3*(B*b^2 - A*b*c)*e)*sqrt(e*x + d))/(b*c^2)]

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Sympy [A]  time = 59.1151, size = 134, normalized size = 1.02 \begin{align*} \frac{2 A d^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} + \frac{2 B \left (d + e x\right )^{\frac{3}{2}}}{3 c} + \frac{\sqrt{d + e x} \left (2 A c e - 2 B b e + 2 B c d\right )}{c^{2}} + \frac{2 \left (- A c + B b\right ) \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^{3} \sqrt{\frac{b e - c d}{c}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(c*x**2+b*x),x)

[Out]

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

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Giac [A]  time = 1.37149, size = 266, normalized size = 2.03 \begin{align*} \frac{2 \, A d^{2} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \,{\left (B b c^{2} d^{2} - A c^{3} d^{2} - 2 \, B b^{2} c d e + 2 \, A b c^{2} d e + B b^{3} e^{2} - A b^{2} c 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^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c^{2} + 3 \, \sqrt{x e + d} B c^{2} d - 3 \, \sqrt{x e + d} B b c e + 3 \, \sqrt{x e + d} A c^{2} e\right )}}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x),x, algorithm="giac")

[Out]

2*A*d^2*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*(B*b*c^2*d^2 - A*c^3*d^2 - 2*B*b^2*c*d*e + 2*A*b*c^2*d
*e + B*b^3*e^2 - A*b^2*c*e^2)*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)*B*c^2 + 3*sqrt(x*e + d)*B*c^2*d - 3*sqrt(x*e + d)*B*b*c*e + 3*sqrt(x*e + d)*A*c^2*e)/c^3