Integrand size = 22, antiderivative size = 146 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {a (a e+c d x)}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\sqrt {a+c x^2}}{c^2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{3/2} e^2}-\frac {d^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \left (c d^2+a e^2\right )^{3/2}} \]
-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
Time = 0.81 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {e \left (2 a^2 e^2+c^2 d^2 x^2+a c \left (d^2+d e x+e^2 x^2\right )\right )}{c^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {2 d^4 \arctan \left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+c x^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+c x^2}}\right )}{c^{3/2}}}{e^2} \]
((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*ArcTan[(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*ArcTan h[(Sqrt[c]*x)/(Sqrt[a] - Sqrt[a + c*x^2])])/c^(3/2))/e^2
Time = 0.46 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {601, 27, 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.
\(\displaystyle \int \frac {x^4}{\left (a+c x^2\right )^{3/2} (d+e x)} \, dx\) |
\(\Big \downarrow \) 601 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\int \frac {a \left (\frac {a d^2}{c d^2+a e^2}-x^2\right )}{c (d+e x) \sqrt {c x^2+a}}dx}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\int \frac {\frac {a d^2}{c d^2+a e^2}-x^2}{(d+e x) \sqrt {c x^2+a}}dx}{c}\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {\int \frac {c d e \left (\frac {a d e}{c d^2+a e^2}+x\right )}{(d+e x) \sqrt {c x^2+a}}dx}{c e^2}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \int \frac {\frac {a d e}{c d^2+a e^2}+x}{(d+e x) \sqrt {c x^2+a}}dx}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \left (\frac {\int \frac {1}{\sqrt {c x^2+a}}dx}{e}-\frac {c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e \left (a e^2+c d^2\right )}\right )}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \left (\frac {\int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{e}-\frac {c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e \left (a e^2+c d^2\right )}\right )}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e}-\frac {c d^3 \int \frac {1}{(d+e x) \sqrt {c x^2+a}}dx}{e \left (a e^2+c d^2\right )}\right )}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \left (\frac {c d^3 \int \frac {1}{c d^2+a e^2-\frac {(a e-c d x)^2}{c x^2+a}}d\frac {a e-c d x}{\sqrt {c x^2+a}}}{e \left (a e^2+c d^2\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e}\right )}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a (a e+c d x)}{c^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {d \left (\frac {c d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e}\right )}{e}-\frac {\sqrt {a+c x^2}}{c e}}{c}\) |
(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*e)) + (d*(ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]]/(Sqrt[c]*e) + (c*d^3* ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e*(c*d^2 + a*e^2)^(3/2))))/e)/c
3.4.34.3.1 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_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && 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. \(350\) 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 | \(\frac {\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}}{e}-\frac {d^{2}}{e^{3} c \sqrt {c \,x^{2}+a}}-\frac {d^{3} x}{e^{4} a \sqrt {c \,x^{2}+a}}-\frac {d \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{e^{2}}+\frac {d^{4} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{5}}\) | \(438\) |
(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)
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 \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[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 ...
\[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Exception generated. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]