3.5.77 \(\int \frac {1}{(d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}})^2} \, dx\) [477]

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

[Out]

2*f^2*(-b^2*f^2+4*a*e^2)*ln(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))/(-b*f^2+2*d*e)^3-2*f^2*(-b^2*f^2+4*a*e^2)*ln(b*
f^2+2*e*(e*x+f*(a+x*(b*f^2+e^2*x)/f^2)^(1/2)))/(-b*f^2+2*d*e)^3-2*(a*e*f^2-b*d*f^2+d^2*e)/(-b*f^2+2*d*e)^2/(d+
e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))-f^2*(-b^2*f^2+4*a*e^2)/(-b*f^2+2*d*e)^2/(b*f^2+2*e*(e*x+f*(a+x*(b*f^2+e^2*x)/
f^2)^(1/2)))

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Rubi [A]
time = 0.16, antiderivative size = 266, normalized size of antiderivative = 1.00, 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 = {2141, 907} \begin {gather*} \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 )} \end {gather*}

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 907

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]))

Rule 2141

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]

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 \text {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 \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(551\) vs. \(2(266)=532\).
time = 3.77, size = 551, normalized size = 2.07 \begin {gather*} \frac {2 \left (b f^3 (-d+e x)+2 e f \left (a f^2-d e x\right )\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}{\left (-2 d e+b f^2\right )^2 \left (d^2+2 d e x-f^2 (a+b x)\right )}-\frac {2 \left (4 a^2 e^3 f^2 x+b d x \left (2 d^2 e^2-2 b d e f^2+b^2 f^4+2 d e^3 x-b e^2 f^2 x\right )+a \left (4 d^3 e^2+2 b e^3 f^2 x^2+d^2 \left (-4 b e f^2+4 e^3 x\right )+d \left (b^2 f^4-6 b e^2 f^2 x-4 e^4 x^2\right )\right )\right )}{(b d-2 a e) \left (-2 d e+b f^2\right )^2 \left (-d^2-2 d e x+f^2 (a+b x)\right )}+\frac {f^2 \left (e+\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (b d f-2 a e f+\sqrt {\frac {e^2}{f^2}} \left (2 d e-b f^2\right ) x+\left (-2 d e+b f^2\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{e \left (2 d e-b f^2\right )^3}-\frac {f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (b f+2 e \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{e \left (2 d e-b f^2\right )^3}+\frac {f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \left (4 a e^2-b^2 f^2\right ) \log \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{e \left (2 d e-b f^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*f^3*(-d + e*x) + 2*e*f*(a*f^2 - d*e*x))*Sqrt[a + x*(b + (e^2*x)/f^2)])/((-2*d*e + b*f^2)^2*(d^2 + 2*d*e*
x - f^2*(a + b*x))) - (2*(4*a^2*e^3*f^2*x + b*d*x*(2*d^2*e^2 - 2*b*d*e*f^2 + b^2*f^4 + 2*d*e^3*x - b*e^2*f^2*x
) + a*(4*d^3*e^2 + 2*b*e^3*f^2*x^2 + d^2*(-4*b*e*f^2 + 4*e^3*x) + d*(b^2*f^4 - 6*b*e^2*f^2*x - 4*e^4*x^2))))/(
(b*d - 2*a*e)*(-2*d*e + b*f^2)^2*(-d^2 - 2*d*e*x + f^2*(a + b*x))) + (f^2*(e + Sqrt[e^2/f^2]*f)*(4*a*e^2 - b^2
*f^2)*Log[b*d*f - 2*a*e*f + Sqrt[e^2/f^2]*(2*d*e - b*f^2)*x + (-2*d*e + b*f^2)*Sqrt[a + x*(b + (e^2*x)/f^2)]])
/(e*(2*d*e - b*f^2)^3) - (f^2*(e - Sqrt[e^2/f^2]*f)*(4*a*e^2 - b^2*f^2)*Log[b*f + 2*e*(-(Sqrt[e^2/f^2]*x) + Sq
rt[a + x*(b + (e^2*x)/f^2)])])/(e*(2*d*e - b*f^2)^3) + (f^2*(e - Sqrt[e^2/f^2]*f)*(4*a*e^2 - b^2*f^2)*Log[d +
f*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + x*(b + (e^2*x)/f^2)])])/(e*(2*d*e - b*f^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58068\) vs. \(2(258)=516\).
time = 0.10, size = 58069, normalized size = 218.30

method	result	size
default	\(\text {Expression too large to display}\)	\(58069\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 803 vs. \(2 (256) = 512\).
time = 3.04, size = 803, normalized size = 3.02 \begin {gather*} -\frac {b^{3} f^{6} x + a b^{2} f^{6} + 3 \, b^{2} d^{2} f^{4} - 16 \, d^{2} x^{2} e^{4} + 16 \, {\left (b d f^{2} x^{2} - d^{3} x\right )} e^{3} - 4 \, {\left (b^{2} f^{4} x^{2} - 2 \, a^{2} f^{4} - 5 \, b d^{2} f^{2} x - 2 \, a d^{2} f^{2}\right )} e^{2} - 2 \, {\left (4 \, b^{2} d f^{4} x + 7 \, a b d f^{4} + b d^{3} f^{2}\right )} e - 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (b^{2} d f^{4} + 8 \, d x^{2} e^{4} - 4 \, {\left (b f^{2} x^{2} + a f^{2} x\right )} e^{3} + 4 \, {\left (2 \, b d f^{2} x + a d f^{2}\right )} e^{2} - {\left (3 \, b^{2} f^{4} x + 4 \, a b f^{4}\right )} e + {\left (b^{2} f^{5} - 4 \, b d f^{3} e - 8 \, d f x e^{3} + 4 \, {\left (b f^{3} x + a f^{3}\right )} e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) - 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (-b f^{2} x - a f^{2} + 2 \, d x e + d^{2}\right ) + 2 \, {\left (b^{3} f^{6} x + a b^{2} f^{6} - 2 \, b^{2} d f^{4} x e - b^{2} d^{2} f^{4} + 8 \, a d f^{2} x e^{3} - 4 \, {\left (a b f^{4} x + a^{2} f^{4} - a d^{2} f^{2}\right )} e^{2}\right )} \log \left (-x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 4 \, {\left (b^{2} d f^{5} - 4 \, d^{2} f x e^{3} + 4 \, {\left (b d f^{3} x + a d f^{3}\right )} e^{2} - {\left (b^{2} f^{5} x + 2 \, a b f^{5} + 2 \, b d^{2} f^{3}\right )} e\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}}{2 \, {\left (b^{4} f^{8} x + a b^{3} f^{8} - b^{3} d^{2} f^{6} + 16 \, d^{4} x e^{4} - 8 \, {\left (4 \, b d^{3} f^{2} x + a d^{3} f^{2} - d^{5}\right )} e^{3} + 12 \, {\left (2 \, b^{2} d^{2} f^{4} x + a b d^{2} f^{4} - b d^{4} f^{2}\right )} e^{2} - 2 \, {\left (4 \, b^{3} d f^{6} x + 3 \, a b^{2} d f^{6} - 3 \, b^{2} d^{3} f^{4}\right )} e\right )}} \end {gather*}

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*(b^3*f^6*x + a*b^2*f^6 + 3*b^2*d^2*f^4 - 16*d^2*x^2*e^4 + 16*(b*d*f^2*x^2 - d^3*x)*e^3 - 4*(b^2*f^4*x^2 -
2*a^2*f^4 - 5*b*d^2*f^2*x - 2*a*d^2*f^2)*e^2 - 2*(4*b^2*d*f^4*x + 7*a*b*d*f^4 + b*d^3*f^2)*e - 2*(b^3*f^6*x +
a*b^2*f^6 - 2*b^2*d*f^4*x*e - b^2*d^2*f^4 + 8*a*d*f^2*x*e^3 - 4*(a*b*f^4*x + a^2*f^4 - a*d^2*f^2)*e^2)*log(b^
2*d*f^4 + 8*d*x^2*e^4 - 4*(b*f^2*x^2 + a*f^2*x)*e^3 + 4*(2*b*d*f^2*x + a*d*f^2)*e^2 - (3*b^2*f^4*x + 4*a*b*f^4
)*e + (b^2*f^5 - 4*b*d*f^3*e - 8*d*f*x*e^3 + 4*(b*f^3*x + a*f^3)*e^2)*sqrt((b*f^2*x + a*f^2 + x^2*e^2)/f^2)) -
2*(b^3*f^6*x + a*b^2*f^6 - 2*b^2*d*f^4*x*e - b^2*d^2*f^4 + 8*a*d*f^2*x*e^3 - 4*(a*b*f^4*x + a^2*f^4 - a*d^2*f
^2)*e^2)*log(-b*f^2*x - a*f^2 + 2*d*x*e + d^2) + 2*(b^3*f^6*x + a*b^2*f^6 - 2*b^2*d*f^4*x*e - b^2*d^2*f^4 + 8*
a*d*f^2*x*e^3 - 4*(a*b*f^4*x + a^2*f^4 - a*d^2*f^2)*e^2)*log(-x*e + f*sqrt((b*f^2*x + a*f^2 + x^2*e^2)/f^2) -
d) - 4*(b^2*d*f^5 - 4*d^2*f*x*e^3 + 4*(b*d*f^3*x + a*d*f^3)*e^2 - (b^2*f^5*x + 2*a*b*f^5 + 2*b*d^2*f^3)*e)*sqr
t((b*f^2*x + a*f^2 + x^2*e^2)/f^2))/(b^4*f^8*x + a*b^3*f^8 - b^3*d^2*f^6 + 16*d^4*x*e^4 - 8*(4*b*d^3*f^2*x + a
*d^3*f^2 - d^5)*e^3 + 12*(2*b^2*d^2*f^4*x + a*b*d^2*f^4 - b*d^4*f^2)*e^2 - 2*(4*b^3*d*f^6*x + 3*a*b^2*d*f^6 -
3*b^2*d^3*f^4)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (256) = 512\).
time = 4.27, size = 1243, normalized size = 4.67 \begin {gather*} \frac {2 \, x e^{2}}{b^{2} f^{4} - 4 \, b d f^{2} e + 4 \, d^{2} e^{2}} + \frac {{\left (b^{3} f^{5} {\left | f \right |} - 2 \, b^{2} d f^{3} {\left | f \right |} e - 4 \, a b f^{3} {\left | f \right |} e^{2} + 8 \, a d f {\left | f \right |} e^{3}\right )} \log \left ({\left | {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b f^{2} - b d f^{2} + 2 \, a f^{2} e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d e \right |}\right )}{b^{4} f^{8} - 8 \, b^{3} d f^{6} e + 24 \, b^{2} d^{2} f^{4} e^{2} - 32 \, b d^{3} f^{2} e^{3} + 16 \, d^{4} e^{4}} + \frac {{\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \log \left ({\left | b f^{2} x + a f^{2} - 2 \, d x e - d^{2} \right |}\right )}{b^{3} f^{6} - 6 \, b^{2} d f^{4} e + 12 \, b d^{2} f^{2} e^{2} - 8 \, d^{3} e^{3}} + \frac {{\left (b^{2} f^{3} {\left | f \right |} e - 4 \, a f {\left | f \right |} e^{3}\right )} \log \left ({\left | b f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right )}{b^{3} f^{6} e - 6 \, b^{2} d f^{4} e^{2} + 12 \, b d^{2} f^{2} e^{3} - 8 \, d^{3} e^{4}} - \frac {{\left (b^{2} f^{3} {\left | f \right |} - 4 \, a f {\left | f \right |} e^{2}\right )} \log \left ({\left | -x e - d + \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} \right |}\right )}{b^{3} f^{6} - 6 \, b^{2} d f^{4} e + 12 \, b d^{2} f^{2} e^{2} - 8 \, d^{3} e^{3}} - \frac {2 \, {\left (b^{3} f^{6} {\left | f \right |} e - 6 \, b^{2} d f^{4} {\left | f \right |} e^{2} + 12 \, b d^{2} f^{2} {\left | f \right |} e^{3} - 8 \, d^{3} {\left | f \right |} e^{4}\right )} \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}}{b^{5} f^{11} - 10 \, b^{4} d f^{9} e + 40 \, b^{3} d^{2} f^{7} e^{2} - 80 \, b^{2} d^{3} f^{5} e^{3} + 80 \, b d^{4} f^{3} e^{4} - 32 \, d^{5} f e^{5}} - \frac {2 \, {\left ({\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b^{3} d f^{6} {\left | f \right |} - {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a b^{2} f^{6} {\left | f \right |} e + a b^{2} d f^{6} {\left | f \right |} e - a^{2} b f^{6} {\left | f \right |} e^{2} - 3 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b^{2} d^{2} f^{4} {\left | f \right |} e + b^{2} d^{3} f^{4} {\left | f \right |} e - 6 \, a b d^{2} f^{4} {\left | f \right |} e^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a^{2} f^{4} {\left | f \right |} e^{3} + 4 \, a^{2} d f^{4} {\left | f \right |} e^{3} + 4 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b d^{3} f^{2} {\left | f \right |} e^{2} - b d^{4} f^{2} {\left | f \right |} e^{2} + 4 \, a d^{3} f^{2} {\left | f \right |} e^{3} - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d^{4} {\left | f \right |} e^{3}\right )}}{{\left (b^{3} f^{7} - 6 \, b^{2} d f^{5} e + 12 \, b d^{2} f^{3} e^{2} - 8 \, d^{3} f e^{3}\right )} {\left ({\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )}^{2} b f^{2} - b d^{2} f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} a f^{2} e + 2 \, a d f^{2} e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )}^{2} d e - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d^{2} e\right )}} - \frac {2 \, {\left (b^{2} d^{2} f^{4} - 2 \, a b d f^{4} e + a^{2} f^{4} e^{2} - 2 \, b d^{3} f^{2} e + 2 \, a d^{2} f^{2} e^{2} + d^{4} e^{2}\right )}}{{\left (b f^{2} x + a f^{2} - 2 \, d x e - d^{2}\right )} {\left (b f^{2} - 2 \, d e\right )}^{3}} \end {gather*}

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]

2*x*e^2/(b^2*f^4 - 4*b*d*f^2*e + 4*d^2*e^2) + (b^3*f^5*abs(f) - 2*b^2*d*f^3*abs(f)*e - 4*a*b*f^3*abs(f)*e^2 +
8*a*d*f*abs(f)*e^3)*log(abs((x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b*f^2 - b*d*f^2 + 2*a*f^2*e - 2*(x*e - sqr
t(b*f^2*x + a*f^2 + x^2*e^2))*d*e))/(b^4*f^8 - 8*b^3*d*f^6*e + 24*b^2*d^2*f^4*e^2 - 32*b*d^3*f^2*e^3 + 16*d^4*
e^4) + (b^2*f^4 - 4*a*f^2*e^2)*log(abs(b*f^2*x + a*f^2 - 2*d*x*e - d^2))/(b^3*f^6 - 6*b^2*d*f^4*e + 12*b*d^2*f
^2*e^2 - 8*d^3*e^3) + (b^2*f^3*abs(f)*e - 4*a*f*abs(f)*e^3)*log(abs(b*f^2 + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^
2*e^2))*e))/(b^3*f^6*e - 6*b^2*d*f^4*e^2 + 12*b*d^2*f^2*e^3 - 8*d^3*e^4) - (b^2*f^3*abs(f) - 4*a*f*abs(f)*e^2)
*log(abs(-x*e - d + sqrt(b*f^2*x + a*f^2 + x^2*e^2)))/(b^3*f^6 - 6*b^2*d*f^4*e + 12*b*d^2*f^2*e^2 - 8*d^3*e^3)
- 2*(b^3*f^6*abs(f)*e - 6*b^2*d*f^4*abs(f)*e^2 + 12*b*d^2*f^2*abs(f)*e^3 - 8*d^3*abs(f)*e^4)*sqrt(b*f^2*x + a
*f^2 + x^2*e^2)/(b^5*f^11 - 10*b^4*d*f^9*e + 40*b^3*d^2*f^7*e^2 - 80*b^2*d^3*f^5*e^3 + 80*b*d^4*f^3*e^4 - 32*d
^5*f*e^5) - 2*((x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b^3*d*f^6*abs(f) - (x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^
2))*a*b^2*f^6*abs(f)*e + a*b^2*d*f^6*abs(f)*e - a^2*b*f^6*abs(f)*e^2 - 3*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2
))*b^2*d^2*f^4*abs(f)*e + b^2*d^3*f^4*abs(f)*e - 6*a*b*d^2*f^4*abs(f)*e^2 + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^
2*e^2))*a^2*f^4*abs(f)*e^3 + 4*a^2*d*f^4*abs(f)*e^3 + 4*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b*d^3*f^2*abs(
f)*e^2 - b*d^4*f^2*abs(f)*e^2 + 4*a*d^3*f^2*abs(f)*e^3 - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*d^4*abs(f)*
e^3)/((b^3*f^7 - 6*b^2*d*f^5*e + 12*b*d^2*f^3*e^2 - 8*d^3*f*e^3)*((x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*b*
f^2 - b*d^2*f^2 + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a*f^2*e + 2*a*d*f^2*e - 2*(x*e - sqrt(b*f^2*x + a*
f^2 + x^2*e^2))^2*d*e - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*d^2*e)) - 2*(b^2*d^2*f^4 - 2*a*b*d*f^4*e + a
^2*f^4*e^2 - 2*b*d^3*f^2*e + 2*a*d^2*f^2*e^2 + d^4*e^2)/((b*f^2*x + a*f^2 - 2*d*x*e - d^2)*(b*f^2 - 2*d*e)^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}\right )}^2} \,d x \end {gather*}

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]

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

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