3.5.78 \(\int \frac {1}{(d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}})^3} \, dx\) [478]

Optimal. Leaf size=330 \[ -\frac {d^2 e-b d f^2+a e f^2}{\left (2 d e-b f^2\right )^2 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^2}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^4}-\frac {6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{\left (2 d e-b f^2\right )^4} \]

[Out]

6*e*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)^4-6*e*f^2*(-b^2*f^2+4*a*e^2)*l
n(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)^4+(-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))^2-2*f^2*(-b^2*f^2+4*a*e^2)/(-b*f^2+2*d*e)^3/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(
1/2))-2*e*f^2*(-b^2*f^2+4*a*e^2)/(-b*f^2+2*d*e)^3/(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 = 0.21, antiderivative size = 330, 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 )}{\left (2 d e-b f^2\right )^3 \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}+\frac {6 e 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 )^4}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \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 {6 e 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 )^4}-\frac {a e f^2-b d f^2+d^2 e}{\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 )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )^3} \, 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^3 \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^3}+\frac {4 a e^2 f^2-b^2 f^4}{\left (2 d e-b f^2\right )^3 x^2}+\frac {3 \left (4 a e^3 f^2-b^2 e f^4\right )}{\left (2 d e-b f^2\right )^4 x}+\frac {2 \left (4 a e^4 f^2-b^2 e^2 f^4\right )}{\left (2 d e-b f^2\right )^3 \left (2 d e-b f^2-2 e x\right )^2}+\frac {6 \left (4 a e^4 f^2-b^2 e^2 f^4\right )}{\left (2 d e-b f^2\right )^4 \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 {d^2 e-b d f^2+a e f^2}{\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 )^2}-\frac {2 f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}-\frac {2 e f^2 \left (4 a e^2-b^2 f^2\right )}{\left (2 d e-b f^2\right )^3 \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 {6 e 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 )^4}-\frac {6 e 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 )^4}\\ \end {align*}

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Mathematica [A]
time = 10.55, size = 300, normalized size = 0.91 \begin {gather*} -\frac {\frac {\left (-2 d e+b f^2\right )^2 \left (d^2 e-b d f^2+a e f^2\right )}{\left (d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )^2}+\frac {2 f^2 \left (-2 d e+b f^2\right ) \left (-4 a e^2+b^2 f^2\right )}{d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}+\frac {2 e f^2 \left (2 d e-b f^2\right ) \left (4 a e^2-b^2 f^2\right )}{b f^2+2 e \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}-6 e f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+6 e 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+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{\left (-2 d e+b f^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(295125\) vs. \(2(320)=640\).
time = 0.14, size = 295126, normalized size = 894.32

method	result	size
default	\(\text {Expression too large to display}\)	\(295126\)

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

[Out]

integrate((x*e + sqrt(b*x + 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. 1942 vs. \(2 (319) = 638\).
time = 26.98, size = 1942, normalized size = 5.88 \begin {gather*} \frac {3 \, b^{4} d f^{8} x + 3 \, a b^{3} d f^{8} - b^{3} d^{3} f^{6} + 32 \, d^{3} x^{3} e^{6} - 8 \, {\left (6 \, b d^{2} f^{2} x^{3} - {\left (a d^{2} f^{2} + 5 \, d^{4}\right )} x^{2}\right )} e^{5} + 8 \, {\left (3 \, b^{2} d f^{4} x^{3} - {\left (a b d f^{4} + 7 \, b d^{3} f^{2}\right )} x^{2} - {\left (7 \, a^{2} d f^{4} + a d^{3} f^{2} - 2 \, d^{5}\right )} x\right )} e^{4} - 2 \, {\left (2 \, b^{3} f^{6} x^{3} - 10 \, a^{3} f^{6} + 12 \, a^{2} d^{2} f^{4} + 6 \, a d^{4} f^{2} - {\left (a b^{2} f^{6} + 11 \, b^{2} d^{2} f^{4}\right )} x^{2} - 2 \, {\left (7 \, a^{2} b f^{6} + 11 \, a b d^{2} f^{4} - 6 \, b d^{4} f^{2}\right )} x\right )} e^{3} - 2 \, {\left (5 \, a^{2} b d f^{6} - 16 \, a b d^{3} f^{4} - b d^{5} f^{2} + 5 \, {\left (a b^{2} d f^{6} - b^{2} d^{3} f^{4}\right )} x\right )} e^{2} - {\left (b^{4} f^{8} x^{2} + 4 \, a^{2} b^{2} f^{8} + 4 \, a b^{2} d^{2} f^{6} + 4 \, b^{2} d^{4} f^{4} + {\left (5 \, a b^{3} f^{8} + 7 \, b^{3} d^{2} f^{6}\right )} x\right )} e + 3 \, {\left (16 \, a d^{2} f^{2} x^{2} e^{5} - 16 \, {\left (a b d f^{4} x^{2} + {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x\right )} e^{4} + 4 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} + {\left (a b^{2} f^{6} - b^{2} d^{2} f^{4}\right )} x^{2} + 2 \, {\left (a^{2} b f^{6} - a b d^{2} f^{4}\right )} x\right )} e^{3} + 4 \, {\left (b^{3} d f^{6} x^{2} + {\left (a b^{2} d f^{6} - b^{2} d^{3} f^{4}\right )} x\right )} e^{2} - {\left (b^{4} f^{8} x^{2} + a^{2} b^{2} f^{8} - 2 \, a b^{2} d^{2} f^{6} + b^{2} d^{4} f^{4} + 2 \, {\left (a b^{3} f^{8} - b^{3} d^{2} f^{6}\right )} x\right )} e\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 ) + 3 \, {\left (16 \, a d^{2} f^{2} x^{2} e^{5} - 16 \, {\left (a b d f^{4} x^{2} + {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x\right )} e^{4} + 4 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} + {\left (a b^{2} f^{6} - b^{2} d^{2} f^{4}\right )} x^{2} + 2 \, {\left (a^{2} b f^{6} - a b d^{2} f^{4}\right )} x\right )} e^{3} + 4 \, {\left (b^{3} d f^{6} x^{2} + {\left (a b^{2} d f^{6} - b^{2} d^{3} f^{4}\right )} x\right )} e^{2} - {\left (b^{4} f^{8} x^{2} + a^{2} b^{2} f^{8} - 2 \, a b^{2} d^{2} f^{6} + b^{2} d^{4} f^{4} + 2 \, {\left (a b^{3} f^{8} - b^{3} d^{2} f^{6}\right )} x\right )} e\right )} \log \left (-b f^{2} x - a f^{2} + 2 \, d x e + d^{2}\right ) - 3 \, {\left (16 \, a d^{2} f^{2} x^{2} e^{5} - 16 \, {\left (a b d f^{4} x^{2} + {\left (a^{2} d f^{4} - a d^{3} f^{2}\right )} x\right )} e^{4} + 4 \, {\left (a^{3} f^{6} - 2 \, a^{2} d^{2} f^{4} + a d^{4} f^{2} + {\left (a b^{2} f^{6} - b^{2} d^{2} f^{4}\right )} x^{2} + 2 \, {\left (a^{2} b f^{6} - a b d^{2} f^{4}\right )} x\right )} e^{3} + 4 \, {\left (b^{3} d f^{6} x^{2} + {\left (a b^{2} d f^{6} - b^{2} d^{3} f^{4}\right )} x\right )} e^{2} - {\left (b^{4} f^{8} x^{2} + a^{2} b^{2} f^{8} - 2 \, a b^{2} d^{2} f^{6} + b^{2} d^{4} f^{4} + 2 \, {\left (a b^{3} f^{8} - b^{3} d^{2} f^{6}\right )} x\right )} e\right )} \log \left (-x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 2 \, {\left (b^{4} f^{9} x + a b^{3} f^{9} + 16 \, d^{3} f x^{2} e^{5} - 12 \, {\left (2 \, b d^{2} f^{3} x^{2} + {\left (3 \, a d^{2} f^{3} - d^{4} f\right )} x\right )} e^{4} + 4 \, {\left (3 \, b^{2} d f^{5} x^{2} + 3 \, a^{2} d f^{5} - 5 \, a d^{3} f^{3} + {\left (9 \, a b d f^{5} - 2 \, b d^{3} f^{3}\right )} x\right )} e^{3} - {\left (2 \, b^{3} f^{7} x^{2} + 6 \, a^{2} b f^{7} - 12 \, a b d^{2} f^{5} - 6 \, b d^{4} f^{3} + 3 \, {\left (3 \, a b^{2} f^{7} - b^{2} d^{2} f^{5}\right )} x\right )} e^{2} - 3 \, {\left (b^{3} d f^{7} x + a b^{2} d f^{7} + b^{2} d^{3} f^{5}\right )} e\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}}{b^{6} f^{12} x^{2} + a^{2} b^{4} f^{12} - 2 \, a b^{4} d^{2} f^{10} + b^{4} d^{4} f^{8} + 64 \, d^{6} x^{2} e^{6} + 2 \, {\left (a b^{5} f^{12} - b^{5} d^{2} f^{10}\right )} x - 64 \, {\left (3 \, b d^{5} f^{2} x^{2} + {\left (a d^{5} f^{2} - d^{7}\right )} x\right )} e^{5} + 16 \, {\left (15 \, b^{2} d^{4} f^{4} x^{2} + a^{2} d^{4} f^{4} - 2 \, a d^{6} f^{2} + d^{8} + 10 \, {\left (a b d^{4} f^{4} - b d^{6} f^{2}\right )} x\right )} e^{4} - 32 \, {\left (5 \, b^{3} d^{3} f^{6} x^{2} + a^{2} b d^{3} f^{6} - 2 \, a b d^{5} f^{4} + b d^{7} f^{2} + 5 \, {\left (a b^{2} d^{3} f^{6} - b^{2} d^{5} f^{4}\right )} x\right )} e^{3} + 4 \, {\left (15 \, b^{4} d^{2} f^{8} x^{2} + 6 \, a^{2} b^{2} d^{2} f^{8} - 12 \, a b^{2} d^{4} f^{6} + 6 \, b^{2} d^{6} f^{4} + 20 \, {\left (a b^{3} d^{2} f^{8} - b^{3} d^{4} f^{6}\right )} x\right )} e^{2} - 4 \, {\left (3 \, b^{5} d f^{10} x^{2} + 2 \, a^{2} b^{3} d f^{10} - 4 \, a b^{3} d^{3} f^{8} + 2 \, b^{3} d^{5} f^{6} + 5 \, {\left (a b^{4} d f^{10} - b^{4} d^{3} f^{8}\right )} x\right )} e} \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))^3,x, algorithm="fricas")

[Out]

(3*b^4*d*f^8*x + 3*a*b^3*d*f^8 - b^3*d^3*f^6 + 32*d^3*x^3*e^6 - 8*(6*b*d^2*f^2*x^3 - (a*d^2*f^2 + 5*d^4)*x^2)*
e^5 + 8*(3*b^2*d*f^4*x^3 - (a*b*d*f^4 + 7*b*d^3*f^2)*x^2 - (7*a^2*d*f^4 + a*d^3*f^2 - 2*d^5)*x)*e^4 - 2*(2*b^3
*f^6*x^3 - 10*a^3*f^6 + 12*a^2*d^2*f^4 + 6*a*d^4*f^2 - (a*b^2*f^6 + 11*b^2*d^2*f^4)*x^2 - 2*(7*a^2*b*f^6 + 11*
a*b*d^2*f^4 - 6*b*d^4*f^2)*x)*e^3 - 2*(5*a^2*b*d*f^6 - 16*a*b*d^3*f^4 - b*d^5*f^2 + 5*(a*b^2*d*f^6 - b^2*d^3*f
^4)*x)*e^2 - (b^4*f^8*x^2 + 4*a^2*b^2*f^8 + 4*a*b^2*d^2*f^6 + 4*b^2*d^4*f^4 + (5*a*b^3*f^8 + 7*b^3*d^2*f^6)*x)
*e + 3*(16*a*d^2*f^2*x^2*e^5 - 16*(a*b*d*f^4*x^2 + (a^2*d*f^4 - a*d^3*f^2)*x)*e^4 + 4*(a^3*f^6 - 2*a^2*d^2*f^4
+ a*d^4*f^2 + (a*b^2*f^6 - b^2*d^2*f^4)*x^2 + 2*(a^2*b*f^6 - a*b*d^2*f^4)*x)*e^3 + 4*(b^3*d*f^6*x^2 + (a*b^2*
d*f^6 - b^2*d^3*f^4)*x)*e^2 - (b^4*f^8*x^2 + a^2*b^2*f^8 - 2*a*b^2*d^2*f^6 + b^2*d^4*f^4 + 2*(a*b^3*f^8 - b^3*
d^2*f^6)*x)*e)*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)) + 3*(16*a*d^2*f^2*x^2*e^5 - 16*(a*b*d*f^4*x^2 + (a^2*d*f^4 - a*d^3*f^2)*x)*e^4 + 4*(a^3*f^
6 - 2*a^2*d^2*f^4 + a*d^4*f^2 + (a*b^2*f^6 - b^2*d^2*f^4)*x^2 + 2*(a^2*b*f^6 - a*b*d^2*f^4)*x)*e^3 + 4*(b^3*d*
f^6*x^2 + (a*b^2*d*f^6 - b^2*d^3*f^4)*x)*e^2 - (b^4*f^8*x^2 + a^2*b^2*f^8 - 2*a*b^2*d^2*f^6 + b^2*d^4*f^4 + 2*
(a*b^3*f^8 - b^3*d^2*f^6)*x)*e)*log(-b*f^2*x - a*f^2 + 2*d*x*e + d^2) - 3*(16*a*d^2*f^2*x^2*e^5 - 16*(a*b*d*f^
4*x^2 + (a^2*d*f^4 - a*d^3*f^2)*x)*e^4 + 4*(a^3*f^6 - 2*a^2*d^2*f^4 + a*d^4*f^2 + (a*b^2*f^6 - b^2*d^2*f^4)*x^
2 + 2*(a^2*b*f^6 - a*b*d^2*f^4)*x)*e^3 + 4*(b^3*d*f^6*x^2 + (a*b^2*d*f^6 - b^2*d^3*f^4)*x)*e^2 - (b^4*f^8*x^2
+ a^2*b^2*f^8 - 2*a*b^2*d^2*f^6 + b^2*d^4*f^4 + 2*(a*b^3*f^8 - b^3*d^2*f^6)*x)*e)*log(-x*e + f*sqrt((b*f^2*x +
a*f^2 + x^2*e^2)/f^2) - d) - 2*(b^4*f^9*x + a*b^3*f^9 + 16*d^3*f*x^2*e^5 - 12*(2*b*d^2*f^3*x^2 + (3*a*d^2*f^3
- d^4*f)*x)*e^4 + 4*(3*b^2*d*f^5*x^2 + 3*a^2*d*f^5 - 5*a*d^3*f^3 + (9*a*b*d*f^5 - 2*b*d^3*f^3)*x)*e^3 - (2*b^
3*f^7*x^2 + 6*a^2*b*f^7 - 12*a*b*d^2*f^5 - 6*b*d^4*f^3 + 3*(3*a*b^2*f^7 - b^2*d^2*f^5)*x)*e^2 - 3*(b^3*d*f^7*x
+ a*b^2*d*f^7 + b^2*d^3*f^5)*e)*sqrt((b*f^2*x + a*f^2 + x^2*e^2)/f^2))/(b^6*f^12*x^2 + a^2*b^4*f^12 - 2*a*b^4
*d^2*f^10 + b^4*d^4*f^8 + 64*d^6*x^2*e^6 + 2*(a*b^5*f^12 - b^5*d^2*f^10)*x - 64*(3*b*d^5*f^2*x^2 + (a*d^5*f^2
- d^7)*x)*e^5 + 16*(15*b^2*d^4*f^4*x^2 + a^2*d^4*f^4 - 2*a*d^6*f^2 + d^8 + 10*(a*b*d^4*f^4 - b*d^6*f^2)*x)*e^4
- 32*(5*b^3*d^3*f^6*x^2 + a^2*b*d^3*f^6 - 2*a*b*d^5*f^4 + b*d^7*f^2 + 5*(a*b^2*d^3*f^6 - b^2*d^5*f^4)*x)*e^3
+ 4*(15*b^4*d^2*f^8*x^2 + 6*a^2*b^2*d^2*f^8 - 12*a*b^2*d^4*f^6 + 6*b^2*d^6*f^4 + 20*(a*b^3*d^2*f^8 - b^3*d^4*f
^6)*x)*e^2 - 4*(3*b^5*d*f^10*x^2 + 2*a^2*b^3*d*f^10 - 4*a*b^3*d^3*f^8 + 2*b^3*d^5*f^6 + 5*(a*b^4*d*f^10 - b^4*
d^3*f^8)*x)*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 )^{3}}\, 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))**3,x)

[Out]

Integral((d + e*x + f*sqrt(a + b*x + 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. 3015 vs. \(2 (319) = 638\).
time = 11.77, size = 3015, normalized size = 9.14 \begin {gather*} \text {Too large to display} \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))^3,x, algorithm="giac")

[Out]

-4*x*e^3/(b^3*f^6 - 6*b^2*d*f^4*e + 12*b*d^2*f^2*e^2 - 8*d^3*e^3) - 3*(b^3*f^5*abs(f)*e - 2*b^2*d*f^3*abs(f)*e
^2 - 4*a*b*f^3*abs(f)*e^3 + 8*a*d*f*abs(f)*e^4)*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 - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*d*e))/(b^5*f^10 - 10*b^4*d*f^8*e + 40*b^3*d^2*f^6*e^
2 - 80*b^2*d^3*f^4*e^3 + 80*b*d^4*f^2*e^4 - 32*d^5*e^5) - 3*(b^2*f^4*e - 4*a*f^2*e^3)*log(abs(-b*f^2*x - a*f^2
+ 2*d*x*e + d^2))/(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) - 3*(b^2*f^3
*abs(f)*e^2 - 4*a*f*abs(f)*e^4)*log(abs(-b*f^2 - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*e))/(b^4*f^8*e - 8*
b^3*d*f^6*e^2 + 24*b^2*d^2*f^4*e^3 - 32*b*d^3*f^2*e^4 + 16*d^4*e^5) + 3*(b^2*f^3*abs(f)*e - 4*a*f*abs(f)*e^3)*
log(abs(-x*e - d + sqrt(b*f^2*x + a*f^2 + x^2*e^2)))/(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) + 4*(b^4*f^8*abs(f)*e^2 - 8*b^3*d*f^6*abs(f)*e^3 + 24*b^2*d^2*f^4*abs(f)*e^4 - 32*b*d^3*
f^2*abs(f)*e^5 + 16*d^4*abs(f)*e^6)*sqrt(b*f^2*x + a*f^2 + x^2*e^2)/(b^7*f^15 - 14*b^6*d*f^13*e + 84*b^5*d^2*f
^11*e^2 - 280*b^4*d^3*f^9*e^3 + 560*b^3*d^4*f^7*e^4 - 672*b^2*d^5*f^5*e^5 + 448*b*d^6*f^3*e^6 - 128*d^7*f*e^7)
+ 2*((x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^3*b^5*f^10*abs(f) + 3*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*
a*b^4*f^10*abs(f)*e + 3*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a^2*b^3*f^10*abs(f)*e^2 - 3*(x*e - sqrt(b*f^2*
x + a*f^2 + x^2*e^2))^3*b^4*d*f^8*abs(f)*e + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*b^4*d^2*f^8*abs(f)*e
- 3*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b^4*d^3*f^8*abs(f)*e - b^4*d^4*f^8*abs(f)*e + a^3*b^2*f^10*abs(f)*
e^3 - 7*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^3*a*b^3*f^8*abs(f)*e^2 - 13*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*
e^2))^2*a*b^3*d*f^8*abs(f)*e^2 + 13*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a*b^3*d^2*f^8*abs(f)*e^2 + a*b^3*d
^3*f^8*abs(f)*e^2 - 16*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a^2*b^2*f^8*abs(f)*e^3 - 26*(x*e - sqrt(b*f^2
*x + a*f^2 + x^2*e^2))*a^2*b^2*d*f^8*abs(f)*e^3 + 5*a^2*b^2*d^2*f^8*abs(f)*e^3 + 3*(x*e - sqrt(b*f^2*x + a*f^2
+ x^2*e^2))^3*b^3*d^2*f^6*abs(f)*e^2 - 11*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*b^3*d^3*f^6*abs(f)*e^2 +
4*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b^3*d^4*f^6*abs(f)*e^2 - b^3*d^5*f^6*abs(f)*e^2 - 8*(x*e - sqrt(b*f^
2*x + a*f^2 + x^2*e^2))*a^3*b*f^8*abs(f)*e^4 - 12*a^3*b*d*f^8*abs(f)*e^4 + 18*(x*e - sqrt(b*f^2*x + a*f^2 + x^
2*e^2))^3*a*b^2*d*f^6*abs(f)*e^3 + 26*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a*b^2*d^2*f^6*abs(f)*e^3 - 14*
(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a*b^2*d^3*f^6*abs(f)*e^3 + 17*a*b^2*d^4*f^6*abs(f)*e^3 + 12*(x*e - sqr
t(b*f^2*x + a*f^2 + x^2*e^2))^3*a^2*b*f^6*abs(f)*e^4 + 40*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a^2*b*d*f^
6*abs(f)*e^4 + 8*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a^2*b*d^2*f^6*abs(f)*e^4 - 44*a^2*b*d^3*f^6*abs(f)*e^
4 - 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^3*b^2*d^3*f^4*abs(f)*e^3 + 14*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*
e^2))^2*b^2*d^4*f^4*abs(f)*e^3 - 8*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b^2*d^5*f^4*abs(f)*e^3 + b^2*d^6*f^
4*abs(f)*e^3 + 20*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a^3*f^6*abs(f)*e^5 + 56*(x*e - sqrt(b*f^2*x + a*f^
2 + x^2*e^2))*a^3*d*f^6*abs(f)*e^5 + 40*a^3*d^2*f^6*abs(f)*e^5 - 12*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^3*
a*b*d^2*f^4*abs(f)*e^4 + 4*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a*b*d^3*f^4*abs(f)*e^4 + 44*(x*e - sqrt(b
*f^2*x + a*f^2 + x^2*e^2))*a*b*d^4*f^4*abs(f)*e^4 - 8*a*b*d^5*f^4*abs(f)*e^4 - 24*(x*e - sqrt(b*f^2*x + a*f^2
+ x^2*e^2))^3*a^2*d*f^4*abs(f)*e^5 - 76*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^2*a^2*d^2*f^4*abs(f)*e^5 - 56*
(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a^2*d^3*f^4*abs(f)*e^5 + 8*a^2*d^4*f^4*abs(f)*e^5 - 12*(x*e - sqrt(b*f
^2*x + a*f^2 + x^2*e^2))^2*b*d^5*f^2*abs(f)*e^4 + 4*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*b*d^6*f^2*abs(f)*e
^4 + 8*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))^3*a*d^3*f^2*abs(f)*e^5 + 4*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^
2))^2*a*d^4*f^2*abs(f)*e^5 - 16*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*a*d^5*f^2*abs(f)*e^5 + 4*(x*e - sqrt(b
*f^2*x + a*f^2 + x^2*e^2))^2*d^6*abs(f)*e^5)/((b^4*f^9 - 8*b^3*d*f^7*e + 24*b^2*d^2*f^5*e^2 - 32*b*d^3*f^3*e^3
+ 16*d^4*f*e^4)*((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) + (3*a*b^3*d*f^8 - 3*a^2*b^2*f^8*e - b^3*d^3*f^6 - 6*a*b^2*d^2*f^6*e - 6*a^2*b*d*f
^6*e^2 - 3*b^2*d^4*f^4*e + 10*a^3*f^6*e^3 + 24*a*b*d^3*f^4*e^2 - 6*a^2*d^2*f^4*e^3 + 6*b*d^5*f^2*e^2 - 18*a*d^
4*f^2*e^3 - 2*d^6*e^3 + 3*(b^4*d*f^8 - a*b^3*f^8*e - 3*b^3*d^2*f^6*e - 2*a*b^2*d*f^6*e^2 + 4*a^2*b*f^6*e^3 + 2
*b^2*d^3*f^4*e^2 + 12*a*b*d^2*f^4*e^3 - 8*a^2*d*f^4*e^4 - 8*a*d^3*f^2*e^4)*x)/((b*f^2*x + a*f^2 - 2*d*x*e - d^
2)^2*(b*f^2 - 2*d*e)^4)

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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 )}^3} \,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))^3,x)

[Out]

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

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