Integrand size = 22, antiderivative size = 152 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \sqrt {c d^2+a e^2}} \]
1/2*(-a*e^2+2*c*d^2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^3+d^3*ar ctanh((-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)^ (1/2)-3/2*d*(c*x^2+a)^(1/2)/c/e^2+1/2*(e*x+d)*(c*x^2+a)^(1/2)/c/e^2
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\frac {e (-2 d+e x) \sqrt {a+c x^2}}{c}+\frac {4 d^3 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {\left (-2 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{2 e^3} \]
((e*(-2*d + e*x)*Sqrt[a + c*x^2])/c + (4*d^3*ArcTan[(Sqrt[c]*(d + e*x) - e *Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]])/Sqrt[-(c*d^2) - a*e^2] + ((-2*c *d^2 + a*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/c^(3/2))/(2*e^3)
Time = 0.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {604, 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.
| \(\displaystyle \int \frac {x^3}{\sqrt {a+c x^2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 604 |
| \(\displaystyle \frac {\int -\frac {3 c d x^2 e^2+a d e^2+\left (c d^2+a e^2\right ) x e}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^3}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {\int \frac {3 c d x^2 e^2+a d e^2+\left (c d^2+a e^2\right ) x e}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {\frac {\int \frac {c e^3 \left (a d e-\left (2 c d^2-a e^2\right ) x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{c e^2}+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \int \frac {a d e-\left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {c x^2+a}}dx+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{e}\right )+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}\right )+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \left (\frac {2 c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}\right )+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \left (-\frac {2 c d^3 \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}\right )+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2}-\frac {e \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{\sqrt {c} e}-\frac {2 c d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \sqrt {a e^2+c d^2}}\right )+3 d e \sqrt {a+c x^2}}{2 c e^3}\) |
((d + e*x)*Sqrt[a + c*x^2])/(2*c*e^2) - (3*d*e*Sqrt[a + c*x^2] + e*(-(((2* c*d^2 - a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*e)) - (2*c*d ^3*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])))/(2*c*e^3)
3.4.27.3.1 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 ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(b*d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b* x^2)^p*ExpandToSum[b*d^m*(m + n + 2*p + 1)*x^m - b*(m + n + 2*p + 1)*(c + d *x)^m - (c + d*x)^(m - 2)*(a*d^2*(m + n - 1) - b*c^2*(m + n + 2*p + 1) - 2* b*c*d*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && IntegerQ[2*p]
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]))
Time = 0.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {\left (-e x +2 d \right ) \sqrt {c \,x^{2}+a}}{2 c \,e^{2}}-\frac {\frac {\left (e^{2} a -2 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{2 e^{2} c}\) | \(201\) |
| default | \(\frac {\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e}+\frac {d^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {d \sqrt {c \,x^{2}+a}}{c \,e^{2}}+\frac {d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(216\) |
-1/2*(-e*x+2*d)*(c*x^2+a)^(1/2)/c/e^2-1/2/e^2/c*((a*e^2-2*c*d^2)/e*ln(x*c^ (1/2)+(c*x^2+a)^(1/2))/c^(1/2)-2*c*d^3/e^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)))
Time = 2.41 (sec) , antiderivative size = 924, normalized size of antiderivative = 6.08 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}\right ] \]
[1/4*(2*sqrt(c*d^2 + a*e^2)*c^2*d^3*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*c^2*d^4 + a*c*d^2*e^2 - a^2*e^4) *sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(2*c^2*d^3*e + 2*a*c*d*e^3 - (c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^3 + a*c^2*e^5), 1/4*(4*sqrt(-c*d^2 - a*e^2)*c^2*d^3*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^4 + a*c*d^2*e^2 - a^2*e^4)*sqrt(c)*log(-2*c*x^2 + 2*sqrt( c*x^2 + a)*sqrt(c)*x - a) - 2*(2*c^2*d^3*e + 2*a*c*d*e^3 - (c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^3 + a*c^2*e^5), 1/2*(sqrt(c*d^2 + a*e^2)*c^2*d^3*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*c^2*d^4 + a*c*d^2*e^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt (-c)*x/sqrt(c*x^2 + a)) - (2*c^2*d^3*e + 2*a*c*d*e^3 - (c^2*d^2*e^2 + a*c* e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^3 + a*c^2*e^5), 1/2*(2*sqrt(-c*d^2 - a *e^2)*c^2*d^3*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^4 + a*c*d^2*e^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (2*c^2*d^3*e + 2*a* c*d*e^3 - (c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^3*d^2*e^3 + a*c^2 *e^5)]
\[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^{3}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Exception generated. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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
Exception generated. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^3}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]