3.850 \(\int \frac{(d+e x)^{3/2}}{\sqrt{f+g x} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=417 \[ -\frac{2 \left (\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{2 \left (e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

[Out]

(2*e^(3/2)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(c*Sqrt[g]) - (2*(e*(2*c*d - b*e) + (2*c^
2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[
d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(c*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]
*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) - (2*(e*(2*c*d - b*e) - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sq
rt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]*Sqrt[f + g*x])])/(c*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
])

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Rubi [A]  time = 3.14052, antiderivative size = 417, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {909, 63, 217, 206, 6728, 93, 208} \[ -\frac{2 \left (\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}+e (2 c d-b e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{c \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{2 \left (e (2 c d-b e)-\frac{-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{c \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 e^{3/2} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a + b*x + c*x^2)),x]

[Out]

(2*e^(3/2)*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(c*Sqrt[g]) - (2*(e*(2*c*d - b*e) + (2*c^
2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[
d + e*x])/(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(c*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]
*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) - (2*(e*(2*c*d - b*e) - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/Sq
rt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]*Sqrt[f + g*x])])/(c*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g
])

Rule 909

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Int
[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(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] && IGtQ[m + 1/2, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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

Mathematica [A]  time = 1.76804, size = 404, normalized size = 0.97 \[ \frac{\left (e \left (b-\sqrt{b^2-4 a c}\right )-2 c d\right )^{3/2} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{g \sqrt{b^2-4 a c}-b g+2 c f}}{\sqrt{f+g x} \sqrt{-e \sqrt{b^2-4 a c}+b e-2 c d}}\right )-\left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )^{3/2} \sqrt{g \left (\sqrt{b^2-4 a c}-b\right )+2 c f} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{e \left (\sqrt{b^2-4 a c}+b\right )-2 c d}}\right )}{c \sqrt{b^2-4 a c} \sqrt{g \left (\sqrt{b^2-4 a c}-b\right )+2 c f} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 (e f-d g)^{3/2} \left (\frac{e (f+g x)}{e f-d g}\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{c \sqrt{g} (f+g x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*f - d*g)^(3/2)*((e*(f + g*x))/(e*f - d*g))^(3/2)*ArcSinh[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/(c*Sq
rt[g]*(f + g*x)^(3/2)) + ((-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*A
rcTan[(Sqrt[2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]*Sqrt[f
+ g*x])] - (-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*Sqrt[2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g]*ArcTan[(Sqrt[2
*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(c
*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Maple [B]  time = 0.695, size = 11688, normalized size = 28. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(c*x^2+b*x+a)/(g*x+f)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )} \sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/((c*x^2 + b*x + a)*sqrt(g*x + f)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError