\(\int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx\) [568]

Optimal result

Integrand size = 19, antiderivative size = 198 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \]

[Out]

-1/2*c^2*d*(-3*a*e^2+2*c*d^2)*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(7/2)-1/
3*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^3-5/6*c*d*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^2-1/6*c*e*(-4*a*
e^2+11*c*d^2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/(e*x+d)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 849, 821, 739, 212} \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=-\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}}-\frac {c e \sqrt {a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {e \sqrt {a+c x^2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]

[In]

Int[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

-1/3*(e*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x)^3) - (5*c*d*e*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^2*(d + e
*x)^2) - (c*e*(11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*(c*d^2 + a*e^2)^3*(d + e*x)) - (c^2*d*(2*c*d^2 - 3*a*e^
2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {c \int \frac {-3 d+2 e x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )} \\ & = -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c \int \frac {2 \left (3 c d^2-2 a e^2\right )-5 c d e x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 \left (c d^2+a e^2\right )^2} \\ & = -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}+\frac {\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^3} \\ & = -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {\left (c^2 d \left (2 c d^2-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^3} \\ & = -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\frac {-e \sqrt {c d^2+a e^2} \sqrt {a+c x^2} \left (2 \left (c d^2+a e^2\right )^2+5 c d \left (c d^2+a e^2\right ) (d+e x)+c \left (11 c d^2-4 a e^2\right ) (d+e x)^2\right )+3 c^2 d \left (2 c d^2-3 a e^2\right ) (d+e x)^3 \log (d+e x)-3 c^2 d \left (2 c d^2-3 a e^2\right ) (d+e x)^3 \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{6 \left (c d^2+a e^2\right )^{7/2} (d+e x)^3} \]

[In]

Integrate[1/((d + e*x)^4*Sqrt[a + c*x^2]),x]

[Out]

(-(e*Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(2*(c*d^2 + a*e^2)^2 + 5*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(11*c*d^2
- 4*a*e^2)*(d + e*x)^2)) + 3*c^2*d*(2*c*d^2 - 3*a*e^2)*(d + e*x)^3*Log[d + e*x] - 3*c^2*d*(2*c*d^2 - 3*a*e^2)*
(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(761\) vs. \(2(178)=356\).

Time = 2.11 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.85

method	result	size
default	\(\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{3 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {5 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{3 \left (e^{2} a +c \,d^{2}\right )}-\frac {2 c \,e^{2} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{3 \left (e^{2} a +c \,d^{2}\right )}}{e^{4}}\)	\(762\)

[In]

int(1/(e*x+d)^4/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-1/3/(a*e^2+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+5/3*c*d*e/(a*e^2
+c*d^2)*(-1/2/(a*e^2+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)+3/2*c*d*e/(a*e
^2+c*d^2)*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-c*d*e/(a*e^2+c*d
^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2
-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2+c*d^2)*e^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(
a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(
1/2))/(x+d/e)))-2/3*c/(a*e^2+c*d^2)*e^2*(-1/(a*e^2+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^
2)/e^2)^(1/2)-c*d*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+
c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (179) = 358\).

Time = 0.62 (sec) , antiderivative size = 1139, normalized size of antiderivative = 5.75 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\left [-\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}, -\frac {3 \, {\left (2 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + {\left (2 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (18 \, c^{3} d^{6} e + 23 \, a c^{2} d^{4} e^{3} + 7 \, a^{2} c d^{2} e^{5} + 2 \, a^{3} e^{7} + {\left (11 \, c^{3} d^{4} e^{3} + 7 \, a c^{2} d^{2} e^{5} - 4 \, a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (9 \, c^{3} d^{5} e^{2} + 8 \, a c^{2} d^{3} e^{4} - a^{2} c d e^{6}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{4} d^{11} + 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} + 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} + {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \, {\left (c^{4} d^{9} e^{2} + 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} + 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \, {\left (c^{4} d^{10} e + 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} + 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}}\right ] \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(2*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*
e^4)*x^2 + 3*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (
2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2
*(18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*
e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 +
6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^
3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e
^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x), -1/6*(3*(2
*c^3*d^6 - 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 - 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 - 3*a*c^2*d^2*e^4)*x^2 + 3
*(2*c^3*d^5*e - 3*a*c^2*d^3*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2
+ a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (18*c^3*d^6*e + 23*a*c^2*d^4*e^3 + 7*a^2*c*d^2*e^5 + 2*a
^3*e^7 + (11*c^3*d^4*e^3 + 7*a*c^2*d^2*e^5 - 4*a^2*c*e^7)*x^2 + 3*(9*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - a^2*c*d*e
^6)*x)*sqrt(c*x^2 + a))/(c^4*d^11 + 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 + 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4
*d^8*e^3 + 4*a*c^3*d^6*e^5 + 6*a^2*c^2*d^4*e^7 + 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 + 4*a*c^3*d^
7*e^4 + 6*a^2*c^2*d^5*e^6 + 4*a^3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e + 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^
6*e^5 + 4*a^3*c*d^4*e^7 + a^4*d^2*e^9)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{4}}\, dx \]

[In]

integrate(1/(e*x+d)**4/(c*x**2+a)**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**2)*(d + e*x)**4), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (179) = 358\).

Time = 0.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.99 \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\frac {1}{3} \, c^{\frac {3}{2}} {\left (\frac {3 \, {\left (2 \, c^{\frac {3}{2}} d^{3} - 3 \, a \sqrt {c} d e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{\frac {3}{2}} d^{3} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a \sqrt {c} d e^{4} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{2} d^{4} e - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c d^{2} e^{3} + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{\frac {5}{2}} d^{5} - 82 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{\frac {3}{2}} d^{3} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} \sqrt {c} d e^{4} - 102 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{2} d^{4} e + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} e^{5} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{\frac {3}{2}} d^{3} e^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} \sqrt {c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} + 4 \, a^{4} e^{5}}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{3}}\right )} \]

[In]

integrate(1/(e*x+d)^4/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/3*c^(3/2)*(3*(2*c^(3/2)*d^3 - 3*a*sqrt(c)*d*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(
-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) - (6*(sqrt(c)*
x - sqrt(c*x^2 + a))^5*c^(3/2)*d^3*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*sqrt(c)*d*e^4 + 30*(sqrt(c)*x - s
qrt(c*x^2 + a))^4*c^2*d^4*e - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + a)
)^3*c^(5/2)*d^5 - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a*c^(3/2)*d^3*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^
2*sqrt(c)*d*e^4 - 102*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^2*d^4*e + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c*d
^2*e^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 - 1
5*(sqrt(c)*x - sqrt(c*x^2 + a))*a^3*sqrt(c)*d*e^4 - 11*a^3*c*d^2*e^3 + 4*a^4*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d -
a*e)^3))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^4 \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^4} \,d x \]

[In]

int(1/((a + c*x^2)^(1/2)*(d + e*x)^4),x)

[Out]

int(1/((a + c*x^2)^(1/2)*(d + e*x)^4), x)