\(\int \frac {x^4}{(d+e x) (a+c x^2)^{3/2}} \, dx\) [334]

Optimal result

Integrand size = 22, antiderivative size = 146 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1661, 1668, 858, 223, 212, 739} \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2}}{c^2 e} \]

[In]

Int[x^4/((d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

(a*(a*e + c*d*x))/(c^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + Sqrt[a + c*x^2]/(c^2*e) - (d*ArcTanh[(Sqrt[c]*x)/Sqr
t[a + c*x^2]])/(c^(3/2)*e^2) - (d^4*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^2*(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 1661

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

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

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {e \left (2 a^2 e^2+c^2 d^2 x^2+a c \left (d^2+d e x+e^2 x^2\right )\right )}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {2 d^4 \arctan \left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+c x^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+c x^2}}\right )}{c^{3/2}}}{e^2} \]

[In]

Integrate[x^4/((d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

((e*(2*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 + d*e*x + e^2*x^2)))/(c^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + (2*d^4*Ar
cTan[(Sqrt[-(c*d^2) - a*e^2]*x)/(Sqrt[a]*(d + e*x) - d*Sqrt[a + c*x^2])])/(-(c*d^2) - a*e^2)^(3/2) + (2*d*ArcT
anh[(Sqrt[c]*x)/(Sqrt[a] - Sqrt[a + c*x^2])])/c^(3/2))/e^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(130)=260\).

Time = 0.47 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.40

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}+a}}{c^{2} e}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}} e^{2}}+\frac {c \,d^{4} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \left (e \sqrt {-a c}+c d \right ) \left (e \sqrt {-a c}-c d \right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {a \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{2 c^{2} \left (e \sqrt {-a c}+c d \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {a \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{2 c^{2} \left (e \sqrt {-a c}-c d \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}\)	\(351\)
default	\(\frac {\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}}{e}-\frac {d^{2}}{e^{3} c \sqrt {c \,x^{2}+a}}-\frac {d^{3} x}{e^{4} a \sqrt {c \,x^{2}+a}}-\frac {d \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{e^{2}}+\frac {d^{4} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{5}}\)	\(438\)

[In]

int(x^4/(e*x+d)/(c*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (131) = 262\).

Time = 3.48 (sec) , antiderivative size = 1525, normalized size of antiderivative = 10.45 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^4/(e*x+d)/(c*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]

[In]

integrate(x**4/(e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

Integral(x**4/((a + c*x**2)**(3/2)*(d + e*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4/(e*x+d)/(c*x^2+a)^(3/2),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 [F(-2)]

Exception generated. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^4/(e*x+d)/(c*x^2+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]

[In]

int(x^4/((a + c*x^2)^(3/2)*(d + e*x)),x)

[Out]

int(x^4/((a + c*x^2)^(3/2)*(d + e*x)), x)