3.477 \(\int \frac{1}{(d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}})^2} \, dx\)

Optimal. Leaf size=266 \[ \frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^3}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]

[Out]

(-2*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])) - (f^2*(4*a*e
^2 - b^2*f^2))/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (2*f^2*(4*a*e^2
- b^2*f^2)*Log[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/(2*d*e - b*f^2)^3 - (2*f^2*(4*a*e^2 - b^2*f^2)*Log
[b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2])])/(2*d*e - b*f^2)^3

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Rubi [A]  time = 0.232046, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2116, 893} \[ \frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^3}-\frac{f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^2 \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac{2 f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt{a+\frac{x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{\left (2 d e-b f^2\right )^3}-\frac{2 \left (a e f^2-b d f^2+d^2 e\right )}{\left (2 d e-b f^2\right )^2 \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])) - (f^2*(4*a*e
^2 - b^2*f^2))/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) + (2*f^2*(4*a*e^2
- b^2*f^2)*Log[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/(2*d*e - b*f^2)^3 - (2*f^2*(4*a*e^2 - b^2*f^2)*Log
[b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2])])/(2*d*e - b*f^2)^3

Rule 2116

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

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.340708, size = 237, normalized size = 0.89 \[ -\frac{\frac{f^2 \left (b^2 f^2-4 a e^2\right ) \left (b f^2-2 d e\right )}{2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )+b f^2}+2 f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x\right )-2 f^2 \left (b^2 f^2-4 a e^2\right ) \log \left (-2 e \left (f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+e x\right )-b f^2\right )+\frac{2 \left (2 d e-b f^2\right ) \left (a e f^2-b d f^2+d^2 e\right )}{f \sqrt{a+x \left (b+\frac{e^2 x}{f^2}\right )}+d+e x}}{\left (2 d e-b f^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 58067, normalized size = 218.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))^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{1}{{\left (e x + \sqrt{b x + \frac{e^{2} x^{2}}{f^{2}} + a} f + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 12.9004, size = 1673, normalized size = 6.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*b^2*f^6 + (3*b^2*d^2 - 14*a*b*d*e + 8*a^2*e^2)*f^4 - 2*(b*d^3*e - 4*a*d^2*e^2)*f^2 - 4*(b^2*e^2*f^4 -
4*b*d*e^3*f^2 + 4*d^2*e^4)*x^2 + (b^3*f^6 - 8*b^2*d*e*f^4 + 20*b*d^2*e^2*f^2 - 16*d^3*e^3)*x - 2*(a*b^2*f^6 +
4*a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4)*x)*log(-
4*a*d*e^2*f^2 - (b^2*d - 4*a*b*e)*f^4 + 4*(b*e^3*f^2 - 2*d*e^4)*x^2 + (3*b^2*e*f^4 - 4*(2*b*d*e^2 - a*e^3)*f^2
)*x - (b^2*f^5 - 4*(b*d*e - a*e^2)*f^3 + 4*(b*e^2*f^3 - 2*d*e^3*f)*x)*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)) -
2*(a*b^2*f^6 + 4*a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 + (b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^
2)*f^4)*x)*log(a*f^2 - d^2 + (b*f^2 - 2*d*e)*x) + 2*(a*b^2*f^6 + 4*a*d^2*e^2*f^2 - (b^2*d^2 + 4*a^2*e^2)*f^4 +
(b^3*f^6 + 8*a*d*e^3*f^2 - 2*(b^2*d*e + 2*a*b*e^2)*f^4)*x)*log(-e*x + f*sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2)
- d) - 4*((b^2*d - 2*a*b*e)*f^5 - 2*(b*d^2*e - 2*a*d*e^2)*f^3 - (b^2*e*f^5 - 4*b*d*e^2*f^3 + 4*d^2*e^3*f)*x)*
sqrt((b*f^2*x + e^2*x^2 + a*f^2)/f^2))/(a*b^3*f^8 + 8*d^5*e^3 - (b^3*d^2 + 6*a*b^2*d*e)*f^6 + 6*(b^2*d^3*e + 2
*a*b*d^2*e^2)*f^4 - 4*(3*b*d^4*e^2 + 2*a*d^3*e^3)*f^2 + (b^4*f^8 - 8*b^3*d*e*f^6 + 24*b^2*d^2*e^2*f^4 - 32*b*d
^3*e^3*f^2 + 16*d^4*e^4)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x + f \sqrt{a + b x + \frac{e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x + f*sqrt(a + b*x + e**2*x**2/f**2))**(-2), x)

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

[Out]

Exception raised: TypeError