\(\int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx\) [344]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a+c x^2}}{c e^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^3}-\frac {d^2 \left (2 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \left (c d^2+a e^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 160, 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 = {1665, 1668, 858, 223, 212, 739} \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {d^2 \left (3 a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3 \left (a e^2+c d^2\right )^{3/2}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^3}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}+\frac {\sqrt {a+c x^2}}{c e^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (a e^2 (d+e x)+c d^2 (2 d+e x)\right )}{c \left (c d^2+a e^2\right ) (d+e x)}+\frac {2 d^2 \left (2 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{e^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(144)=288\).

Time = 0.44 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.41

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}+a}}{c \,e^{2}}-\frac {2 d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}+\frac {d^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{e^{3} \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {d^{4} c \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {3 d^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(386\)
default	\(\frac {\sqrt {c \,x^{2}+a}}{c \,e^{2}}-\frac {2 d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {d^{3} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{5}}-\frac {3 d^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\)	\(390\)

[In]

int(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (145) = 290\).

Time = 5.64 (sec) , antiderivative size = 1449, normalized size of antiderivative = 9.06 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]

[In]

integrate(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(x**3/(e*x+d)**2/(c*x**2+a)**(1/2),x)

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(x^3/(e*x+d)^2/(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 [F]

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

[In]

integrate(x^3/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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