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

Optimal. Leaf size=198 \[ -\frac {e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^3}-\frac {5 c d e \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c e \left (11 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{6 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{7/2}} \]

[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)

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Rubi [A]
time = 0.10, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 849, 821, 739, 212} \begin {gather*} -\frac {c^2 d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\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 )} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {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*}

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Mathematica [A]
time = 10.15, size = 209, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(761\) vs. \(2(178)=356\).
time = 0.45, size = 762, normalized size = 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}}\)	\(762\)

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (179) = 358\).
time = 0.34, size = 466, normalized size = 2.35 \begin {gather*} \frac {5 \, c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {5 \, \sqrt {c x^{2} + a} c^{2} d^{2}}{2 \, {\left (c^{3} d^{7} e^{\left (-1\right )} + c^{3} d^{6} x + 3 \, a c^{2} d^{4} x e^{2} + 3 \, a c^{2} d^{5} e + 3 \, a^{2} c d^{2} x e^{4} + 3 \, a^{2} c d^{3} e^{3} + a^{3} x e^{6} + a^{3} d e^{5}\right )}} - \frac {3 \, c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}}} - \frac {5 \, \sqrt {c x^{2} + a} c d}{6 \, {\left (c^{2} d^{4} x^{2} e + c^{2} d^{6} e^{\left (-1\right )} + 2 \, c^{2} d^{5} x + 2 \, a c d^{2} x^{2} e^{3} + 4 \, a c d^{3} x e^{2} + 2 \, a c d^{4} e + a^{2} x^{2} e^{5} + 2 \, a^{2} d x e^{4} + a^{2} d^{2} e^{3}\right )}} + \frac {2 \, \sqrt {c x^{2} + a} c}{3 \, {\left (c^{2} d^{5} e^{\left (-1\right )} + c^{2} d^{4} x + 2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )}} - \frac {\sqrt {c x^{2} + a}}{3 \, {\left (c d^{2} x^{3} e^{2} + 3 \, c d^{3} x^{2} e + c d^{5} e^{\left (-1\right )} + 3 \, c d^{4} x + a x^{3} e^{4} + 3 \, a d x^{2} e^{3} + 3 \, a d^{2} x e^{2} + a d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*c^3*d^3*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(sqrt(a*c)*abs(x*e + d)))*e^(-7)/(c*d^2*e^(-2) + a)^(
7/2) - 5/2*sqrt(c*x^2 + a)*c^2*d^2/(c^3*d^7*e^(-1) + c^3*d^6*x + 3*a*c^2*d^4*x*e^2 + 3*a*c^2*d^5*e + 3*a^2*c*d
^2*x*e^4 + 3*a^2*c*d^3*e^3 + a^3*x*e^6 + a^3*d*e^5) - 3/2*c^2*d*arcsinh(c*d*x/(sqrt(a*c)*abs(x*e + d)) - a*e/(
sqrt(a*c)*abs(x*e + d)))*e^(-5)/(c*d^2*e^(-2) + a)^(5/2) - 5/6*sqrt(c*x^2 + a)*c*d/(c^2*d^4*x^2*e + c^2*d^6*e^
(-1) + 2*c^2*d^5*x + 2*a*c*d^2*x^2*e^3 + 4*a*c*d^3*x*e^2 + 2*a*c*d^4*e + a^2*x^2*e^5 + 2*a^2*d*x*e^4 + a^2*d^2
*e^3) + 2/3*sqrt(c*x^2 + a)*c/(c^2*d^5*e^(-1) + c^2*d^4*x + 2*a*c*d^2*x*e^2 + 2*a*c*d^3*e + a^2*x*e^4 + a^2*d*
e^3) - 1/3*sqrt(c*x^2 + a)/(c*d^2*x^3*e^2 + 3*c*d^3*x^2*e + c*d^5*e^(-1) + 3*c*d^4*x + a*x^3*e^4 + 3*a*d*x^2*e
^3 + 3*a*d^2*x*e^2 + a*d^3*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (179) = 358\).
time = 3.24, size = 1134, normalized size = 5.73 \begin {gather*} \left [-\frac {3 \, {\left (6 \, c^{3} d^{5} x e + 2 \, c^{3} d^{6} - 3 \, a c^{2} d x^{3} e^{5} - 9 \, a c^{2} d^{2} x^{2} e^{4} + {\left (2 \, c^{3} d^{3} x^{3} - 9 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} - a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (27 \, c^{3} d^{5} x e^{2} + 18 \, c^{3} d^{6} e + 24 \, a c^{2} d^{3} x e^{4} - 3 \, a^{2} c d x e^{6} - 2 \, {\left (2 \, a^{2} c x^{2} - a^{3}\right )} e^{7} + 7 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{5} + {\left (11 \, c^{3} d^{4} x^{2} + 23 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (3 \, c^{4} d^{10} x e + c^{4} d^{11} + a^{4} x^{3} e^{11} + 3 \, a^{4} d x^{2} e^{10} + {\left (4 \, a^{3} c d^{2} x^{3} + 3 \, a^{4} d^{2} x\right )} e^{9} + {\left (12 \, a^{3} c d^{3} x^{2} + a^{4} d^{3}\right )} e^{8} + 6 \, {\left (a^{2} c^{2} d^{4} x^{3} + 2 \, a^{3} c d^{4} x\right )} e^{7} + 2 \, {\left (9 \, a^{2} c^{2} d^{5} x^{2} + 2 \, a^{3} c d^{5}\right )} e^{6} + 2 \, {\left (2 \, a c^{3} d^{6} x^{3} + 9 \, a^{2} c^{2} d^{6} x\right )} e^{5} + 6 \, {\left (2 \, a c^{3} d^{7} x^{2} + a^{2} c^{2} d^{7}\right )} e^{4} + {\left (c^{4} d^{8} x^{3} + 12 \, a c^{3} d^{8} x\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + 4 \, a c^{3} d^{9}\right )} e^{2}\right )}}, \frac {3 \, {\left (6 \, c^{3} d^{5} x e + 2 \, c^{3} d^{6} - 3 \, a c^{2} d x^{3} e^{5} - 9 \, a c^{2} d^{2} x^{2} e^{4} + {\left (2 \, c^{3} d^{3} x^{3} - 9 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} - a c^{2} d^{4}\right )} e^{2}\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}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) - {\left (27 \, c^{3} d^{5} x e^{2} + 18 \, c^{3} d^{6} e + 24 \, a c^{2} d^{3} x e^{4} - 3 \, a^{2} c d x e^{6} - 2 \, {\left (2 \, a^{2} c x^{2} - a^{3}\right )} e^{7} + 7 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{5} + {\left (11 \, c^{3} d^{4} x^{2} + 23 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (3 \, c^{4} d^{10} x e + c^{4} d^{11} + a^{4} x^{3} e^{11} + 3 \, a^{4} d x^{2} e^{10} + {\left (4 \, a^{3} c d^{2} x^{3} + 3 \, a^{4} d^{2} x\right )} e^{9} + {\left (12 \, a^{3} c d^{3} x^{2} + a^{4} d^{3}\right )} e^{8} + 6 \, {\left (a^{2} c^{2} d^{4} x^{3} + 2 \, a^{3} c d^{4} x\right )} e^{7} + 2 \, {\left (9 \, a^{2} c^{2} d^{5} x^{2} + 2 \, a^{3} c d^{5}\right )} e^{6} + 2 \, {\left (2 \, a c^{3} d^{6} x^{3} + 9 \, a^{2} c^{2} d^{6} x\right )} e^{5} + 6 \, {\left (2 \, a c^{3} d^{7} x^{2} + a^{2} c^{2} d^{7}\right )} e^{4} + {\left (c^{4} d^{8} x^{3} + 12 \, a c^{3} d^{8} x\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + 4 \, a c^{3} d^{9}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/12*(3*(6*c^3*d^5*x*e + 2*c^3*d^6 - 3*a*c^2*d*x^3*e^5 - 9*a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3 - 9*a*c^2*d^3*
x)*e^3 + 3*(2*c^3*d^4*x^2 - a*c^2*d^4)*e^2)*sqrt(c*d^2 + a*e^2)*log(-(2*c^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 -
2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a) + (a*c*x^2 + 2*a^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(2
7*c^3*d^5*x*e^2 + 18*c^3*d^6*e + 24*a*c^2*d^3*x*e^4 - 3*a^2*c*d*x*e^6 - 2*(2*a^2*c*x^2 - a^3)*e^7 + 7*(a*c^2*d
^2*x^2 + a^2*c*d^2)*e^5 + (11*c^3*d^4*x^2 + 23*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c^4*d^10*x*e + c^4*d^11 + a
^4*x^3*e^11 + 3*a^4*d*x^2*e^10 + (4*a^3*c*d^2*x^3 + 3*a^4*d^2*x)*e^9 + (12*a^3*c*d^3*x^2 + a^4*d^3)*e^8 + 6*(a
^2*c^2*d^4*x^3 + 2*a^3*c*d^4*x)*e^7 + 2*(9*a^2*c^2*d^5*x^2 + 2*a^3*c*d^5)*e^6 + 2*(2*a*c^3*d^6*x^3 + 9*a^2*c^2
*d^6*x)*e^5 + 6*(2*a*c^3*d^7*x^2 + a^2*c^2*d^7)*e^4 + (c^4*d^8*x^3 + 12*a*c^3*d^8*x)*e^3 + (3*c^4*d^9*x^2 + 4*
a*c^3*d^9)*e^2), 1/6*(3*(6*c^3*d^5*x*e + 2*c^3*d^6 - 3*a*c^2*d*x^3*e^5 - 9*a*c^2*d^2*x^2*e^4 + (2*c^3*d^3*x^3
- 9*a*c^2*d^3*x)*e^3 + 3*(2*c^3*d^4*x^2 - a*c^2*d^4)*e^2)*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)/(c^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + a^2)*e^2)) - (27*c^3*d^5*x*e^2 + 18*c^3*d^6*e
+ 24*a*c^2*d^3*x*e^4 - 3*a^2*c*d*x*e^6 - 2*(2*a^2*c*x^2 - a^3)*e^7 + 7*(a*c^2*d^2*x^2 + a^2*c*d^2)*e^5 + (11*c
^3*d^4*x^2 + 23*a*c^2*d^4)*e^3)*sqrt(c*x^2 + a))/(3*c^4*d^10*x*e + c^4*d^11 + a^4*x^3*e^11 + 3*a^4*d*x^2*e^10
+ (4*a^3*c*d^2*x^3 + 3*a^4*d^2*x)*e^9 + (12*a^3*c*d^3*x^2 + a^4*d^3)*e^8 + 6*(a^2*c^2*d^4*x^3 + 2*a^3*c*d^4*x)
*e^7 + 2*(9*a^2*c^2*d^5*x^2 + 2*a^3*c*d^5)*e^6 + 2*(2*a*c^3*d^6*x^3 + 9*a^2*c^2*d^6*x)*e^5 + 6*(2*a*c^3*d^7*x^
2 + a^2*c^2*d^7)*e^4 + (c^4*d^8*x^3 + 12*a*c^3*d^8*x)*e^3 + (3*c^4*d^9*x^2 + 4*a*c^3*d^9)*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (179) = 358\).
time = 0.64, size = 578, normalized size = 2.92 \begin {gather*} \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 {30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{2} d^{4} e + 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{\frac {5}{2}} d^{5} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{\frac {3}{2}} d^{3} e^{2} - 102 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{2} d^{4} e - 82 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{\frac {3}{2}} d^{3} e^{2} - 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c d^{2} e^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a \sqrt {c} d e^{4} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{\frac {3}{2}} d^{3} e^{2} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c d^{2} e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} \sqrt {c} d e^{4} - 11 \, a^{3} c d^{2} e^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} \sqrt {c} d e^{4} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} e^{5} + 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)) - (30*(sqrt(c)
*x - sqrt(c*x^2 + a))^4*c^2*d^4*e + 44*(sqrt(c)*x - sqrt(c*x^2 + a))^3*c^(5/2)*d^5 + 6*(sqrt(c)*x - sqrt(c*x^2
+ a))^5*c^(3/2)*d^3*e^2 - 102*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a*c^2*d^4*e - 82*(sqrt(c)*x - sqrt(c*x^2 + a))^
3*a*c^(3/2)*d^3*e^2 - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*sqr
t(c)*d*e^4 + 60*(sqrt(c)*x - sqrt(c*x^2 + a))*a^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^2*c*d
^2*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^2*sqrt(c)*d*e^4 - 11*a^3*c*d^2*e^3 - 15*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^3*sqrt(c)*d*e^4 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^3*e^5 + 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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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