\(\int (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}})^3 \, dx\) [454]

Optimal result

Integrand size = 25, antiderivative size = 175 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=-\frac {a d^3 f^2}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {a d f^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}{4 e}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}+\frac {3 a d^2 f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \]

[Out]

3/2*a*d^2*f^2*ln(e*x+f*(a+e^2*x^2/f^2)^(1/2))/e-1/2*a*d^3*f^2/e/(e*x+f*(a+e^2*x^2/f^2)^(1/2))+a*d*f^2*(e*x+f*(
a+e^2*x^2/f^2)^(1/2))/e+1/4*a*f^2*(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^2/e+1/8*(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^4/e

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2142, 907} \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=-\frac {a d^3 f^2}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}+\frac {3 a d^2 f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac {\left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac {a f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac {a d f^2 \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{e} \]

[In]

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

[Out]

-1/2*(a*d^3*f^2)/(e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (a*d*f^2*(e*x + f*Sqrt[a + (e^2*x^2)/f^2]))/e + (a*f^
2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2)/(4*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^4/(8*e) + (3*a*d^2*f^
2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(2*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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3 \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \left (2 a d f^2+\frac {a d^3 f^2}{(d-x)^2}-\frac {3 a d^2 f^2}{d-x}+a f^2 x+x^3\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ & = -\frac {a d^3 f^2}{2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}+\frac {a d f^2 \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e}+\frac {a f^2 \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2}{4 e}+\frac {\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^4}{8 e}+\frac {3 a d^2 f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {e x \left (2 d^3+6 a d f^2+3 d^2 e x+3 a e f^2 x+4 d e^2 x^2+2 e^3 x^3\right )+\sqrt {a+\frac {e^2 x^2}{f^2}} \left (2 a f^3 (2 d+e x)+e f x \left (3 d^2+4 d e x+2 e^2 x^2\right )\right )-3 a d^2 f^2 \log \left (e \left (\sqrt {a} f+e x-f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )+3 a d^2 f^2 \log \left (-\sqrt {a} f+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(157)=314\).

Time = 0.93 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.95

method	result	size
default	\(f^{3} \left (\frac {x \left (a +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{4}\right )+3 f^{2} \left (\frac {e^{3} x^{4}}{4 f^{2}}+\frac {d \,e^{2} x^{3}}{3 f^{2}}+\frac {a e \,x^{2}}{2}+a d x \right )+3 f \left (d^{2} \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )+e^{2} \left (\frac {x \left (a +\frac {e^{2} x^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{2}}{4 e^{2}}-\frac {a \,f^{2} \left (\frac {x \sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}}{2}+\frac {a \ln \left (\frac {e^{2} x}{f^{2} \sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {a +\frac {e^{2} x^{2}}{f^{2}}}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}\right )}{4 e^{2}}\right )+\frac {2 d \,f^{2} \left (\frac {e^{2} x^{2}+a \,f^{2}}{f^{2}}\right )^{\frac {3}{2}}}{3 e}\right )+\frac {\left (e x +d \right )^{4}}{4 e}\)	\(341\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.92 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 3 \, a d^{2} f^{2} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \, {\left (a e^{2} f^{2} + d^{2} e^{2}\right )} x^{2} + 2 \, {\left (3 \, a d e f^{2} + d^{3} e\right )} x + {\left (2 \, e^{3} f x^{3} + 4 \, d e^{2} f x^{2} + 4 \, a d f^{3} + {\left (2 \, a e f^{3} + 3 \, d^{2} e f\right )} x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \, e} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.62 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=3 a d f^{2} x + \frac {3 a e f^{2} x^{2}}{2} + a f^{3} \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}{2} & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + d^{3} x + \frac {3 d^{2} e x^{2}}{2} + 3 d^{2} f \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}}{2} & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + 2 d e^{2} x^{3} + 6 d e f \left (\begin {cases} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \left (\frac {a f^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + e^{3} x^{4} + 4 e^{2} f \left (\begin {cases} - \frac {a^{2} f^{2} \left (\begin {cases} \frac {\log {\left (\frac {2 e^{2} x}{f^{2}} + 2 \sqrt {\frac {e^{2}}{f^{2}}} \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \right )}}{\sqrt {\frac {e^{2}}{f^{2}}}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {\frac {e^{2} x^{2}}{f^{2}}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}} \left (\frac {a f^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: \frac {e^{2}}{f^{2}} \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

3*a*d*f**2*x + 3*a*e*f**2*x**2/2 + a*f**3*Piecewise((a*Piecewise((log(2*e**2*x/f**2 + 2*sqrt(e**2/f**2)*sqrt(a
+ e**2*x**2/f**2))/sqrt(e**2/f**2), Ne(a, 0)), (x*log(x)/sqrt(e**2*x**2/f**2), True))/2 + x*sqrt(a + e**2*x**
2/f**2)/2, Ne(e**2/f**2, 0)), (sqrt(a)*x, True)) + d**3*x + 3*d**2*e*x**2/2 + 3*d**2*f*Piecewise((a*Piecewise(
(log(2*e**2*x/f**2 + 2*sqrt(e**2/f**2)*sqrt(a + e**2*x**2/f**2))/sqrt(e**2/f**2), Ne(a, 0)), (x*log(x)/sqrt(e*
*2*x**2/f**2), True))/2 + x*sqrt(a + e**2*x**2/f**2)/2, Ne(e**2/f**2, 0)), (sqrt(a)*x, True)) + 2*d*e**2*x**3
+ 6*d*e*f*Piecewise((sqrt(a + e**2*x**2/f**2)*(a*f**2/(3*e**2) + x**2/3), Ne(e**2/f**2, 0)), (sqrt(a)*x**2/2,
True)) + e**3*x**4 + 4*e**2*f*Piecewise((-a**2*f**2*Piecewise((log(2*e**2*x/f**2 + 2*sqrt(e**2/f**2)*sqrt(a +
e**2*x**2/f**2))/sqrt(e**2/f**2), Ne(a, 0)), (x*log(x)/sqrt(e**2*x**2/f**2), True))/(8*e**2) + sqrt(a + e**2*x
**2/f**2)*(a*f**2*x/(8*e**2) + x**3/4), Ne(e**2/f**2, 0)), (sqrt(a)*x**3/3, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.67 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\frac {1}{4} \, e^{3} x^{4} + \frac {3 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{2} f^{4}}{4 \, e} - \frac {3}{8} \, {\left (\frac {a^{2} f^{3} \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}} e^{2}} - \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{2} x}{e^{2}} + \frac {\sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a f^{2} x}{e^{2}}\right )} e^{2} f + \frac {1}{8} \, {\left (\frac {3 \, a^{2} f \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}}} + 2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} x + 3 \, \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} a x\right )} f^{3} + d^{3} x + \frac {3}{2} \, {\left (e x^{2} + {\left (\frac {a f \operatorname {arsinh}\left (\frac {e^{2} x}{\sqrt {a e^{2}} f}\right )}{\sqrt {e^{2}}} + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} x\right )} f\right )} d^{2} + {\left (e^{2} x^{3} + \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{3}}{e} + {\left (\frac {e^{2} x^{3}}{f^{2}} + 3 \, a x\right )} f^{2}\right )} d \]

[In]

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

[Out]

1/4*e^3*x^4 + 3/4*(e^2*x^2/f^2 + a)^2*f^4/e - 3/8*(a^2*f^3*arcsinh(e^2*x/(sqrt(a*e^2)*f))/(sqrt(e^2)*e^2) - 2*
(e^2*x^2/f^2 + a)^(3/2)*f^2*x/e^2 + sqrt(e^2*x^2/f^2 + a)*a*f^2*x/e^2)*e^2*f + 1/8*(3*a^2*f*arcsinh(e^2*x/(sqr
t(a*e^2)*f))/sqrt(e^2) + 2*(e^2*x^2/f^2 + a)^(3/2)*x + 3*sqrt(e^2*x^2/f^2 + a)*a*x)*f^3 + d^3*x + 3/2*(e*x^2 +
(a*f*arcsinh(e^2*x/(sqrt(a*e^2)*f))/sqrt(e^2) + sqrt(e^2*x^2/f^2 + a)*x)*f)*d^2 + (e^2*x^3 + 2*(e^2*x^2/f^2 +
a)^(3/2)*f^3/e + (e^2*x^3/f^2 + 3*a*x)*f^2)*d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.98 \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=e^{3} x^{4} + \frac {3}{2} \, a e f^{2} x^{2} + 2 \, d e^{2} x^{3} + 3 \, a d f^{2} x + \frac {3}{2} \, d^{2} e x^{2} - \frac {3 \, a d^{2} f {\left | f \right |} \log \left ({\left | -x {\left | e \right |} + \sqrt {e^{2} x^{2} + a f^{2}} \right |}\right )}{2 \, {\left | e \right |}} + d^{3} x + \frac {1}{2} \, \sqrt {e^{2} x^{2} + a f^{2}} {\left (\frac {4 \, a d f {\left | f \right |}}{e} + {\left (2 \, {\left (\frac {e^{2} x {\left | f \right |}}{f} + \frac {2 \, d e {\left | f \right |}}{f}\right )} x + \frac {2 \, a e^{4} f^{4} {\left | f \right |} + 3 \, d^{2} e^{4} f^{2} {\left | f \right |}}{e^{4} f^{3}}\right )} x\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^3 \, dx=\int {\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^3 \,d x \]

[In]

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

[Out]

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