Integrand size = 22, antiderivative size = 211 \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=-\frac {\left (8 c d^3-e \left (4 c d^2-a e^2\right ) x\right ) \sqrt {a+c x^2}}{8 c e^4}-\frac {7 d \left (a+c x^2\right )^{3/2}}{12 c e^2}+\frac {(d+e x) \left (a+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {d^3 \sqrt {c d^2+a e^2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^5} \]
-7/12*d*(c*x^2+a)^(3/2)/c/e^2+1/4*(e*x+d)*(c*x^2+a)^(3/2)/c/e^2+1/8*(-a^2* e^4+4*a*c*d^2*e^2+8*c^2*d^4)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(3/2)/e^ 5+d^3*arctanh((-c*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))*(a*e^2+c*d ^2)^(1/2)/e^5-1/8*(8*c*d^3-e*(-a*e^2+4*c*d^2)*x)*(c*x^2+a)^(1/2)/c/e^4
Time = 0.44 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\frac {\sqrt {c} e \sqrt {a+c x^2} \left (a e^2 (-8 d+3 e x)+c \left (-24 d^3+12 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )\right )-48 c^{3/2} d^3 \sqrt {-c d^2-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )-3 \left (8 c^2 d^4+4 a c d^2 e^2-a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{24 c^{3/2} e^5} \]
(Sqrt[c]*e*Sqrt[a + c*x^2]*(a*e^2*(-8*d + 3*e*x) + c*(-24*d^3 + 12*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3)) - 48*c^(3/2)*d^3*Sqrt[-(c*d^2) - a*e^2]*ArcTa n[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]] - 3*(8*c ^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(24 *c^(3/2)*e^5)
Time = 0.55 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {604, 25, 2185, 27, 682, 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 |
| \(\displaystyle \frac {\int -\frac {\sqrt {c x^2+a} \left (7 c d x^2 e^2+a d e^2+\left (3 c d^2+a e^2\right ) x e\right )}{d+e x}dx}{4 c e^3}+\frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {\int \frac {\sqrt {c x^2+a} \left (7 c d x^2 e^2+a d e^2+\left (3 c d^2+a e^2\right ) x e\right )}{d+e x}dx}{4 c e^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {\frac {\int \frac {3 c e^3 \left (a d e-\left (4 c d^2-a e^2\right ) x\right ) \sqrt {c x^2+a}}{d+e x}dx}{3 c e^2}+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \int \frac {\left (a d e-\left (4 c d^2-a e^2\right ) x\right ) \sqrt {c x^2+a}}{d+e x}dx+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {\int \frac {c \left (a d e \left (4 c d^2+a e^2\right )-\left (8 c^2 d^4+4 a c e^2 d^2-a^2 e^4\right ) x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{2 c e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {\int \frac {a d e \left (4 c d^2+a e^2\right )-\left (8 c^2 d^4+4 a c e^2 d^2-a^2 e^4\right ) x}{(d+e x) \sqrt {c x^2+a}}dx}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {\frac {8 c d^3 \left (a e^2+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{e}}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {\frac {8 c d^3 \left (a e^2+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {\frac {8 c d^3 \left (a e^2+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e}}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {-\frac {8 c d^3 \left (a e^2+c d^2\right ) \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e}}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+c x^2\right )^{3/2} (d+e x)}{4 c e^2}-\frac {e \left (\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (-a^2 e^4+4 a c d^2 e^2+8 c^2 d^4\right )}{\sqrt {c} e}-\frac {8 c d^3 \sqrt {a e^2+c d^2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e}}{2 e^2}+\frac {\sqrt {a+c x^2} \left (8 c d^3-e x \left (4 c d^2-a e^2\right )\right )}{2 e^2}\right )+\frac {7}{3} d e \left (a+c x^2\right )^{3/2}}{4 c e^3}\) |
((d + e*x)*(a + c*x^2)^(3/2))/(4*c*e^2) - ((7*d*e*(a + c*x^2)^(3/2))/3 + e *(((8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*Sqrt[a + c*x^2])/(2*e^2) + (-(((8*c^2 *d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqr t[c]*e)) - (8*c*d^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e)/(2*e^2)))/(4*c*e^3)
3.4.17.3.1 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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 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 = 0.44 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(-\frac {\left (-6 c \,x^{3} e^{3}+8 c d \,e^{2} x^{2}-3 a \,e^{3} x -12 c \,d^{2} e x +8 a d \,e^{2}+24 c \,d^{3}\right ) \sqrt {c \,x^{2}+a}}{24 c \,e^{4}}-\frac {\frac {\left (a^{2} e^{4}-4 a c \,d^{2} e^{2}-8 c^{2} d^{4}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {8 d^{3} \left (e^{2} a +c \,d^{2}\right ) c \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{8 c \,e^{4}}\) | \(266\) |
| default | \(\frac {\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 c}-\frac {a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4 c}}{e}+\frac {d^{2} \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{e^{3}}-\frac {d \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 c \,e^{2}}-\frac {d^{3} \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\) | \(387\) |
-1/24*(-6*c*e^3*x^3+8*c*d*e^2*x^2-3*a*e^3*x-12*c*d^2*e*x+8*a*d*e^2+24*c*d^ 3)*(c*x^2+a)^(1/2)/c/e^4-1/8/c/e^4*((a^2*e^4-4*a*c*d^2*e^2-8*c^2*d^4)/e*ln (x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-8*d^3*(a*e^2+c*d^2)*c/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 = 4.13 (sec) , antiderivative size = 963, normalized size of antiderivative = 4.56 \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\left [\frac {24 \, \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 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 (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{2} e^{5}}, \frac {48 \, \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 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 (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{48 \, c^{2} e^{5}}, \frac {12 \, \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 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 (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{2} e^{5}}, \frac {24 \, \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 ) - 3 \, {\left (8 \, c^{2} d^{4} + 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 (6 \, c^{2} e^{4} x^{3} - 8 \, c^{2} d e^{3} x^{2} - 24 \, c^{2} d^{3} e - 8 \, a c d e^{3} + 3 \, {\left (4 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{24 \, c^{2} e^{5}}\right ] \]
[1/48*(24*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)) - 3*(8*c^2*d^4 + 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*(6*c^2* e^4*x^3 - 8*c^2*d*e^3*x^2 - 24*c^2*d^3*e - 8*a*c*d*e^3 + 3*(4*c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^2*e^5), 1/48*(48*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)) - 3*(8*c^2*d^4 + 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*(6*c^2*e^ 4*x^3 - 8*c^2*d*e^3*x^2 - 24*c^2*d^3*e - 8*a*c*d*e^3 + 3*(4*c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^2*e^5), 1/24*(12*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)) - 3*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/s qrt(c*x^2 + a)) + (6*c^2*e^4*x^3 - 8*c^2*d*e^3*x^2 - 24*c^2*d^3*e - 8*a*c* d*e^3 + 3*(4*c^2*d^2*e^2 + a*c*e^4)*x)*sqrt(c*x^2 + a))/(c^2*e^5), 1/24*(2 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)*s qrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 3*(8*c^2*d ^4 + 4*a*c*d^2*e^2 - a^2*e^4)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (6*c^2*e^4*x^3 - 8*c^2*d*e^3*x^2 - 24*c^2*d^3*e - 8*a*c*d*e^3 + 3*(4*...
\[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\int \frac {x^{3} \sqrt {a + c x^{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 \sqrt {a+c x^2}}{d+e x} \, dx=\int \frac {x^3\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \]