3.4.43 \(\int \frac {x^4}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [343]

Optimal. Leaf size=204 \[ -\frac {5 d \sqrt {a+c x^2}}{2 c e^3}-\frac {d^4 \sqrt {a+c x^2}}{e^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^3}+\frac {\left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac {d^3 \left (3 c d^2+4 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^4 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1665, 1668, 858, 223, 212, 739} \begin {gather*} \frac {\left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac {d^4 \sqrt {a+c x^2}}{e^3 (d+e x) \left (a e^2+c d^2\right )}+\frac {d^3 \left (4 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac {5 d \sqrt {a+c x^2}}{2 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

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

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Mathematica [A]
time = 1.03, size = 210, normalized size = 1.03 \begin {gather*} \frac {\frac {e \sqrt {a+c x^2} \left (c d^2 \left (-6 d^2-3 d e x+e^2 x^2\right )+a e^2 \left (-4 d^2-3 d e x+e^2 x^2\right )\right )}{c \left (c d^2+a e^2\right ) (d+e x)}-\frac {4 d^3 \left (3 c d^2+4 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {\left (-6 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(180)=360\).
time = 0.09, size = 435, normalized size = 2.13

method	result	size
risch	\(-\frac {\left (-e x +4 d \right ) \sqrt {c \,x^{2}+a}}{2 c \,e^{3}}-\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) a}{2 e^{2} c^{\frac {3}{2}}}+\frac {3 d^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}+\frac {4 d^{3} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {d^{4} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{e^{4} \left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c \,d^{5} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\)	\(424\)
default	\(\frac {\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e^{2}}-\frac {2 d \sqrt {c \,x^{2}+a}}{c \,e^{3}}+\frac {3 d^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{4} \sqrt {c}}+\frac {4 d^{3} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {d^{4} \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{e^{6}}\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*(1/2*x/c*(c*x^2+a)^(1/2)-1/2*a/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2)))-2*d*(c*x^2+a)^(1/2)/c/e^3+3*d^2/e^
4*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)+4/e^5*d^3/((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/e^6*d^4*(-
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 [A]
time = 0.32, size = 227, normalized size = 1.11 \begin {gather*} \frac {c d^{5} \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 )}}{{\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {4 \, 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 (-5\right )}}{\sqrt {c d^{2} e^{\left (-2\right )} + a}} - \frac {\sqrt {c x^{2} + a} d^{4}}{c d^{2} x e^{4} + c d^{3} e^{3} + a x e^{6} + a d e^{5}} + \frac {3 \, d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-4\right )}}{\sqrt {c}} + \frac {\sqrt {c x^{2} + a} x e^{\left (-2\right )}}{2 \, c} - \frac {a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{2 \, c^{\frac {3}{2}}} - \frac {2 \, \sqrt {c x^{2} + a} d e^{\left (-3\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (174) = 348\).
time = 78.12, size = 1723, normalized size = 8.45 \begin {gather*} \left [-\frac {{\left (6 \, c^{3} d^{6} x e + 6 \, c^{3} d^{7} + 11 \, a c^{2} d^{4} x e^{3} + 11 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{2} x e^{5} + 4 \, a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (3 \, c^{3} d^{5} x e + 3 \, c^{3} d^{6} + 4 \, a c^{2} d^{3} x e^{3} + 4 \, a c^{2} d^{4} 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 (3 \, c^{3} d^{5} x e^{2} + 6 \, c^{3} d^{6} e + 6 \, a c^{2} d^{3} x e^{4} - a^{2} c x^{2} e^{7} + 3 \, a^{2} c d x e^{6} - 2 \, {\left (a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{5} - {\left (c^{3} d^{4} x^{2} - 10 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{4} d^{4} x e^{5} + c^{4} d^{5} e^{4} + 2 \, a c^{3} d^{2} x e^{7} + 2 \, a c^{3} d^{3} e^{6} + a^{2} c^{2} x e^{9} + a^{2} c^{2} d e^{8}\right )}}, -\frac {4 \, {\left (3 \, c^{3} d^{5} x e + 3 \, c^{3} d^{6} + 4 \, a c^{2} d^{3} x e^{3} + 4 \, a c^{2} d^{4} 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 (6 \, c^{3} d^{6} x e + 6 \, c^{3} d^{7} + 11 \, a c^{2} d^{4} x e^{3} + 11 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{2} x e^{5} + 4 \, a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, c^{3} d^{5} x e^{2} + 6 \, c^{3} d^{6} e + 6 \, a c^{2} d^{3} x e^{4} - a^{2} c x^{2} e^{7} + 3 \, a^{2} c d x e^{6} - 2 \, {\left (a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{5} - {\left (c^{3} d^{4} x^{2} - 10 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{4} d^{4} x e^{5} + c^{4} d^{5} e^{4} + 2 \, a c^{3} d^{2} x e^{7} + 2 \, a c^{3} d^{3} e^{6} + a^{2} c^{2} x e^{9} + a^{2} c^{2} d e^{8}\right )}}, -\frac {{\left (6 \, c^{3} d^{6} x e + 6 \, c^{3} d^{7} + 11 \, a c^{2} d^{4} x e^{3} + 11 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{2} x e^{5} + 4 \, a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, c^{3} d^{5} x e + 3 \, c^{3} d^{6} + 4 \, a c^{2} d^{3} x e^{3} + 4 \, a c^{2} d^{4} 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 ) + {\left (3 \, c^{3} d^{5} x e^{2} + 6 \, c^{3} d^{6} e + 6 \, a c^{2} d^{3} x e^{4} - a^{2} c x^{2} e^{7} + 3 \, a^{2} c d x e^{6} - 2 \, {\left (a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{5} - {\left (c^{3} d^{4} x^{2} - 10 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{4} d^{4} x e^{5} + c^{4} d^{5} e^{4} + 2 \, a c^{3} d^{2} x e^{7} + 2 \, a c^{3} d^{3} e^{6} + a^{2} c^{2} x e^{9} + a^{2} c^{2} d e^{8}\right )}}, -\frac {2 \, {\left (3 \, c^{3} d^{5} x e + 3 \, c^{3} d^{6} + 4 \, a c^{2} d^{3} x e^{3} + 4 \, a c^{2} d^{4} 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 (6 \, c^{3} d^{6} x e + 6 \, c^{3} d^{7} + 11 \, a c^{2} d^{4} x e^{3} + 11 \, a c^{2} d^{5} e^{2} + 4 \, a^{2} c d^{2} x e^{5} + 4 \, a^{2} c d^{3} e^{4} - a^{3} x e^{7} - a^{3} d e^{6}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (3 \, c^{3} d^{5} x e^{2} + 6 \, c^{3} d^{6} e + 6 \, a c^{2} d^{3} x e^{4} - a^{2} c x^{2} e^{7} + 3 \, a^{2} c d x e^{6} - 2 \, {\left (a c^{2} d^{2} x^{2} - 2 \, a^{2} c d^{2}\right )} e^{5} - {\left (c^{3} d^{4} x^{2} - 10 \, a c^{2} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{4} d^{4} x e^{5} + c^{4} d^{5} e^{4} + 2 \, a c^{3} d^{2} x e^{7} + 2 \, a c^{3} d^{3} e^{6} + a^{2} c^{2} x e^{9} + a^{2} c^{2} d e^{8}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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