3.5.59 \(\int \frac {1}{(d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}})^3} \, dx\) [459]

Optimal. Leaf size=193 \[ -\frac {a f^2}{2 d^3 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {1+\frac {a f^2}{d^2}}{4 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}-\frac {a f^2}{d^3 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {3 a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2142, 907} \begin {gather*} -\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{2 d^4 e}-\frac {a f^2}{2 d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {a f^2}{d^3 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{4 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*f^2)/(d^3*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(4*e*(d + e*x + f*Sqrt[a + (e^2*x^2
)/f^2])^2) - (a*f^2)/(d^3*e*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (3*a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f
^2]])/(2*d^4*e) + (3*a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*d^4*e)

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 2142

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[(g + h*x^n)^p*((d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, 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+\frac {e^2 x^2}{f^2}}\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {d^2+a f^2-2 d x+x^2}{(d-x)^2 x^3} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a f^2}{d^3 (d-x)^2}+\frac {3 a f^2}{d^4 (d-x)}+\frac {d^2+a f^2}{d^2 x^3}+\frac {2 a f^2}{d^3 x^2}+\frac {3 a f^2}{d^4 x}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac {a f^2}{2 d^3 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {1+\frac {a f^2}{d^2}}{4 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}-\frac {a f^2}{d^3 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {3 a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e}+\frac {3 a f^2 \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 d^4 e}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(464\) vs. \(2(193)=386\).
time = 1.91, size = 464, normalized size = 2.40 \begin {gather*} \frac {-\frac {2 d \sqrt {a+\frac {e^2 x^2}{f^2}} \left (3 a^2 f^5+d^2 e f x (3 d+4 e x)-a d f^3 (5 d+9 e x)\right )}{e \left (d^2-a f^2+2 d e x\right )^2}+\frac {2 d x \left (2 d^8+3 a^4 f^8+5 d^7 e x-3 a d^5 e f^2 x+15 a^2 d^3 e f^4 x-9 a^3 d e f^6 x+a d^4 f^2 \left (3 a f^2-8 e^2 x^2\right )+a^2 d^2 f^4 \left (-9 a f^2+4 e^2 x^2\right )+d^6 \left (a f^2+4 e^2 x^2\right )\right )}{\left (d^2-a f^2\right )^2 \left (d^2-a f^2+2 d e x\right )^2}-\frac {3 a f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \log \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e^2}+\frac {3 a f \log \left (a f+d \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{\sqrt {\frac {e^2}{f^2}}}+\frac {3 a f^2 \log \left (d^3 e \left (a f+d \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )\right )}{e}+\frac {3 a f^2 \left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \log \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{e^2}}{4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*d*Sqrt[a + (e^2*x^2)/f^2]*(3*a^2*f^5 + d^2*e*f*x*(3*d + 4*e*x) - a*d*f^3*(5*d + 9*e*x)))/(e*(d^2 - a*f^2
+ 2*d*e*x)^2) + (2*d*x*(2*d^8 + 3*a^4*f^8 + 5*d^7*e*x - 3*a*d^5*e*f^2*x + 15*a^2*d^3*e*f^4*x - 9*a^3*d*e*f^6*x
+ a*d^4*f^2*(3*a*f^2 - 8*e^2*x^2) + a^2*d^2*f^4*(-9*a*f^2 + 4*e^2*x^2) + d^6*(a*f^2 + 4*e^2*x^2)))/((d^2 - a*
f^2)^2*(d^2 - a*f^2 + 2*d*e*x)^2) - (3*a*f^2*(e - Sqrt[e^2/f^2]*f)*Log[-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)
/f^2]])/e^2 + (3*a*f*Log[a*f + d*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2])])/Sqrt[e^2/f^2] + (3*a*f^2*Log
[d^3*e*(a*f + d*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2]))])/e + (3*a*f^2*(e - Sqrt[e^2/f^2]*f)*Log[d + f
*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2])])/e^2)/(4*d^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(9870\) vs. \(2(176)=352\).
time = 0.06, size = 9871, normalized size = 51.15

method	result	size
default	\(\text {Expression too large to display}\)	\(9871\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d+e*x+f*(a+1/f^2*e^2*x^2)^(1/2))^3,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+e^2*x^2/f^2)^(1/2))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (170) = 340\).
time = 0.41, size = 521, normalized size = 2.70 \begin {gather*} \frac {5 \, a^{3} f^{6} - 6 \, a^{2} d^{2} f^{4} - 3 \, a d^{4} f^{2} + 8 \, d^{3} x^{3} e^{3} + 2 \, {\left (a d^{2} f^{2} + 5 \, d^{4}\right )} x^{2} e^{2} - 2 \, {\left (7 \, a^{2} d f^{4} + a d^{3} f^{2} - 2 \, d^{5}\right )} x e + 3 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + 4 \, a d^{2} f^{2} x^{2} e^{2} + a d^{4} f^{2} - 4 \, {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x e\right )} \log \left (-a f^{2} x e + a d f^{2} + 2 \, d x^{2} e^{2} + {\left (a f^{3} - 2 \, d f x e\right )} \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) + 3 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + 4 \, a d^{2} f^{2} x^{2} e^{2} + a d^{4} f^{2} - 4 \, {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x e\right )} \log \left (-a f^{2} + 2 \, d x e + d^{2}\right ) - 3 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + 4 \, a d^{2} f^{2} x^{2} e^{2} + a d^{4} f^{2} - 4 \, {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x e\right )} \log \left (-x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 2 \, {\left (3 \, a^{2} d f^{5} - 5 \, a d^{3} f^{3} + 4 \, d^{3} f x^{2} e^{2} - 3 \, {\left (3 \, a d^{2} f^{3} - d^{4} f\right )} x e\right )} \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}}}{4 \, {\left (4 \, d^{6} x^{2} e^{3} - 4 \, {\left (a d^{5} f^{2} - d^{7}\right )} x e^{2} + {\left (a^{2} d^{4} f^{4} - 2 \, a d^{6} f^{2} + d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(5*a^3*f^6 - 6*a^2*d^2*f^4 - 3*a*d^4*f^2 + 8*d^3*x^3*e^3 + 2*(a*d^2*f^2 + 5*d^4)*x^2*e^2 - 2*(7*a^2*d*f^4
+ a*d^3*f^2 - 2*d^5)*x*e + 3*(a^3*f^6 - 2*a^2*d^2*f^4 + 4*a*d^2*f^2*x^2*e^2 + a*d^4*f^2 - 4*(a^2*d*f^4 - a*d^3
*f^2)*x*e)*log(-a*f^2*x*e + a*d*f^2 + 2*d*x^2*e^2 + (a*f^3 - 2*d*f*x*e)*sqrt((a*f^2 + x^2*e^2)/f^2)) + 3*(a^3*
f^6 - 2*a^2*d^2*f^4 + 4*a*d^2*f^2*x^2*e^2 + a*d^4*f^2 - 4*(a^2*d*f^4 - a*d^3*f^2)*x*e)*log(-a*f^2 + 2*d*x*e +
d^2) - 3*(a^3*f^6 - 2*a^2*d^2*f^4 + 4*a*d^2*f^2*x^2*e^2 + a*d^4*f^2 - 4*(a^2*d*f^4 - a*d^3*f^2)*x*e)*log(-x*e
+ f*sqrt((a*f^2 + x^2*e^2)/f^2) - d) - 2*(3*a^2*d*f^5 - 5*a*d^3*f^3 + 4*d^3*f*x^2*e^2 - 3*(3*a*d^2*f^3 - d^4*f
)*x*e)*sqrt((a*f^2 + x^2*e^2)/f^2))/(4*d^6*x^2*e^3 - 4*(a*d^5*f^2 - d^7)*x*e^2 + (a^2*d^4*f^4 - 2*a*d^6*f^2 +
d^8)*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 + \frac {e^{2} x^{2}}{f^{2}}}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (170) = 340\).
time = 4.98, size = 679, normalized size = 3.52 \begin {gather*} \frac {3 \, a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | a f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d \right |}\right )}{4 \, d^{4}} + \frac {3 \, a f^{2} e^{\left (-1\right )} \log \left ({\left | -a f^{2} + 2 \, d x e + d^{2} \right |}\right )}{4 \, d^{4}} - \frac {3 \, a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e - d + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{4 \, d^{4}} + \frac {3 \, a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{4 \, d^{4}} - \frac {\sqrt {a f^{2} + x^{2} e^{2}} {\left | f \right |} e^{\left (-1\right )}}{2 \, d^{3} f} + \frac {x}{2 \, d^{3}} + \frac {{\left (5 \, a^{3} f^{6} - 3 \, a^{2} d^{2} f^{4} - 9 \, a d^{4} f^{2} - d^{6} - 12 \, {\left (a^{2} d f^{4} e + a d^{3} f^{2} e\right )} x\right )} e^{\left (-1\right )}}{8 \, {\left (a f^{2} - 2 \, d x e - d^{2}\right )}^{2} d^{4}} + \frac {{\left (5 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} a^{3} f^{6} {\left | f \right |} + 14 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a^{3} d f^{6} {\left | f \right |} + 10 \, a^{3} d^{2} f^{6} {\left | f \right |} - 6 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{3} a^{2} d f^{4} {\left | f \right |} - 19 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} a^{2} d^{2} f^{4} {\left | f \right |} - 14 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a^{2} d^{3} f^{4} {\left | f \right |} + 2 \, a^{2} d^{4} f^{4} {\left | f \right |} + 2 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{3} a d^{3} f^{2} {\left | f \right |} + {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} a d^{4} f^{2} {\left | f \right |} - 4 \, {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a d^{5} f^{2} {\left | f \right |} + {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} d^{6} {\left | f \right |}\right )} e^{\left (-1\right )}}{8 \, {\left ({\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a f^{2} + a d f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} d - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d^{2}\right )}^{2} d^{4} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a*f*abs(f)*e^(-1)*log(abs(a*f^2 - (x*e - sqrt(a*f^2 + x^2*e^2))*d))/d^4 + 3/4*a*f^2*e^(-1)*log(abs(-a*f^2
+ 2*d*x*e + d^2))/d^4 - 3/4*a*f*abs(f)*e^(-1)*log(abs(-x*e - d + sqrt(a*f^2 + x^2*e^2)))/d^4 + 3/4*a*f*abs(f)*
e^(-1)*log(abs(-x*e + sqrt(a*f^2 + x^2*e^2)))/d^4 - 1/2*sqrt(a*f^2 + x^2*e^2)*abs(f)*e^(-1)/(d^3*f) + 1/2*x/d^
3 + 1/8*(5*a^3*f^6 - 3*a^2*d^2*f^4 - 9*a*d^4*f^2 - d^6 - 12*(a^2*d*f^4*e + a*d^3*f^2*e)*x)*e^(-1)/((a*f^2 - 2*
d*x*e - d^2)^2*d^4) + 1/8*(5*(x*e - sqrt(a*f^2 + x^2*e^2))^2*a^3*f^6*abs(f) + 14*(x*e - sqrt(a*f^2 + x^2*e^2))
*a^3*d*f^6*abs(f) + 10*a^3*d^2*f^6*abs(f) - 6*(x*e - sqrt(a*f^2 + x^2*e^2))^3*a^2*d*f^4*abs(f) - 19*(x*e - sqr
t(a*f^2 + x^2*e^2))^2*a^2*d^2*f^4*abs(f) - 14*(x*e - sqrt(a*f^2 + x^2*e^2))*a^2*d^3*f^4*abs(f) + 2*a^2*d^4*f^4
*abs(f) + 2*(x*e - sqrt(a*f^2 + x^2*e^2))^3*a*d^3*f^2*abs(f) + (x*e - sqrt(a*f^2 + x^2*e^2))^2*a*d^4*f^2*abs(f
) - 4*(x*e - sqrt(a*f^2 + x^2*e^2))*a*d^5*f^2*abs(f) + (x*e - sqrt(a*f^2 + x^2*e^2))^2*d^6*abs(f))*e^(-1)/(((x
*e - sqrt(a*f^2 + x^2*e^2))*a*f^2 + a*d*f^2 - (x*e - sqrt(a*f^2 + x^2*e^2))^2*d - (x*e - sqrt(a*f^2 + x^2*e^2)
)*d^2)^2*d^4*f)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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