3.1777 \(\int \frac{(a+b x)^2}{(c+d x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]

[Out]

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

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Rubi [A]  time = 0.117035, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {88, 63, 208} \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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

Mathematica [A]  time = 0.0897442, size = 112, normalized size = 1. \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.009, size = 201, normalized size = 1.8 \begin{align*}{\frac{2\,{b}^{2}}{3\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+4\,{\frac{ab\sqrt{fx+e}}{df}}-2\,{\frac{{b}^{2}c\sqrt{fx+e}}{f{d}^{2}}}-2\,{\frac{{b}^{2}e\sqrt{fx+e}}{d{f}^{2}}}+2\,{\frac{{a}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43694, size = 779, normalized size = 6.96 \begin{align*} \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{d^{2} e - c d f} f^{2} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (2 \, b^{2} d^{3} e^{2} +{\left (b^{2} c d^{2} - 6 \, a b d^{3}\right )} e f - 3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} f^{2} -{\left (b^{2} d^{3} e f - b^{2} c d^{2} f^{2}\right )} x\right )} \sqrt{f x + e}}{3 \,{\left (d^{4} e f^{2} - c d^{3} f^{3}\right )}}, \frac{2 \,{\left (3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-d^{2} e + c d f} f^{2} \arctan \left (\frac{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}{d f x + d e}\right ) -{\left (2 \, b^{2} d^{3} e^{2} +{\left (b^{2} c d^{2} - 6 \, a b d^{3}\right )} e f - 3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} f^{2} -{\left (b^{2} d^{3} e f - b^{2} c d^{2} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{3 \,{\left (d^{4} e f^{2} - c d^{3} f^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(d^2*e - c*d*f)*f^2*log((d*f*x + 2*d*e - c*f - 2*sqrt(d^2*e - c*d*
f)*sqrt(f*x + e))/(d*x + c)) - 2*(2*b^2*d^3*e^2 + (b^2*c*d^2 - 6*a*b*d^3)*e*f - 3*(b^2*c^2*d - 2*a*b*c*d^2)*f^
2 - (b^2*d^3*e*f - b^2*c*d^2*f^2)*x)*sqrt(f*x + e))/(d^4*e*f^2 - c*d^3*f^3), 2/3*(3*(b^2*c^2 - 2*a*b*c*d + a^2
*d^2)*sqrt(-d^2*e + c*d*f)*f^2*arctan(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e)/(d*f*x + d*e)) - (2*b^2*d^3*e^2 + (b^
2*c*d^2 - 6*a*b*d^3)*e*f - 3*(b^2*c^2*d - 2*a*b*c*d^2)*f^2 - (b^2*d^3*e*f - b^2*c*d^2*f^2)*x)*sqrt(f*x + e))/(
d^4*e*f^2 - c*d^3*f^3)]

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Sympy [A]  time = 22.1489, size = 110, normalized size = 0.98 \begin{align*} \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}}}{3 d f^{2}} + \frac{2 b \sqrt{e + f x} \left (2 a d f - b c f - b d e\right )}{d^{2} f^{2}} - \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{d^{2} \sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*(e + f*x)**(3/2)/(3*d*f**2) + 2*b*sqrt(e + f*x)*(2*a*d*f - b*c*f - b*d*e)/(d**2*f**2) - 2*(a*d - b*c)**
2*atan(1/(sqrt(d/(c*f - d*e))*sqrt(e + f*x)))/(d**2*sqrt(d/(c*f - d*e))*(c*f - d*e))

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Giac [A]  time = 1.49055, size = 203, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{2}} + \frac{2 \,{\left ({\left (f x + e\right )}^{\frac{3}{2}} b^{2} d^{2} f^{4} - 3 \, \sqrt{f x + e} b^{2} c d f^{5} + 6 \, \sqrt{f x + e} a b d^{2} f^{5} - 3 \, \sqrt{f x + e} b^{2} d^{2} f^{4} e\right )}}{3 \, d^{3} f^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d^2) + 2/3*
((f*x + e)^(3/2)*b^2*d^2*f^4 - 3*sqrt(f*x + e)*b^2*c*d*f^5 + 6*sqrt(f*x + e)*a*b*d^2*f^5 - 3*sqrt(f*x + e)*b^2
*d^2*f^4*e)/(d^3*f^6)