Integrand size = 22, antiderivative size = 160 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a+c x^2}}{c e^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^3}-\frac {d^2 \left (2 c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \left (c d^2+a e^2\right )^{3/2}} \]
-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*arctanh(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 )
Time = 0.80 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2} \left (a e^2 (d+e x)+c d^2 (2 d+e x)\right )}{c \left (c d^2+a e^2\right ) (d+e x)}+\frac {2 d^2 \left (2 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 d \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{e^3} \]
((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
Time = 0.51 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {603, 25, 2185, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3}{\sqrt {a+c x^2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 603 |
| \(\displaystyle \frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {\int -\frac {\frac {a d^2}{e}-\left (\frac {c d^2}{e^2}+a\right ) x d+\frac {\left (c d^2+a e^2\right ) x^2}{e}}{(d+e x) \sqrt {c x^2+a}}dx}{a e^2+c d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {a d^2}{e}-\left (\frac {c d^2}{e^2}+a\right ) x d+\frac {\left (c d^2+a e^2\right ) x^2}{e}}{(d+e x) \sqrt {c x^2+a}}dx}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\int \frac {c d \left (a d e-2 \left (c d^2+a e^2\right ) x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{c e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {a d e-2 \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt {c x^2+a}}dx}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (3 a e^2+2 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {2 \left (a e^2+c d^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{e}\right )}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (3 a e^2+2 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {2 \left (a e^2+c d^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}\right )}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {d \left (\frac {d \left (3 a e^2+2 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}\right )}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {d \left (-\frac {d \left (3 a e^2+2 c d^2\right ) \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}\right )}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {d \left (-\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (a e^2+c d^2\right )}{\sqrt {c} e}-\frac {d \left (3 a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \sqrt {a e^2+c d^2}}\right )}{e^2}+\sqrt {a+c x^2} \left (\frac {a}{c}+\frac {d^2}{e^2}\right )}{a e^2+c d^2}+\frac {d^3 \sqrt {a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}\) |
(d^3*Sqrt[a + c*x^2])/(e^2*(c*d^2 + a*e^2)*(d + e*x)) + ((a/c + d^2/e^2)*S qrt[a + c*x^2] + (d*((-2*(c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^ 2]])/(Sqrt[c]*e) - (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])])/(e*Sqrt[c*d^2 + a*e^2])))/e^2)/(c*d^2 + a*e^ 2)
3.4.44.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x) ^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(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] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(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 + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
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 | \(\frac {\sqrt {c \,x^{2}+a}}{c \,e^{2}}-\frac {2 d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {d^{3} \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{5}}-\frac {3 d^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(390\) |
(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))
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 \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]
[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*s qrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + ...
\[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^{3}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {x^{3}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x^3}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]