Integrand size = 19, antiderivative size = 154 \[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {(d+e x)^{1+m} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \operatorname {AppellF1}\left (1+m,\frac {3}{2},\frac {3}{2},2+m,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m) \left (a+c x^2\right )^{3/2}} \]
(e*x+d)^(1+m)*AppellF1(1+m,3/2,3/2,2+m,(e*x+d)/(d-e*(-a)^(1/2)/c^(1/2)),(e *x+d)/(d+e*(-a)^(1/2)/c^(1/2)))*(1+(-e*x-d)/(d-e*(-a)^(1/2)/c^(1/2)))^(3/2 )*(1+(-e*x-d)/(d+e*(-a)^(1/2)/c^(1/2)))^(3/2)/e/(1+m)/(c*x^2+a)^(3/2)
Time = 0.55 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{3/2} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}\right )^{3/2} (d+e x)^{1+m} \operatorname {AppellF1}\left (1+m,\frac {3}{2},\frac {3}{2},2+m,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (1+m) \left (a+c x^2\right )^{3/2}} \]
(((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^(3/2)*((e*(Sqrt[-(a/c)] + x ))/(-d + Sqrt[-(a/c)]*e))^(3/2)*(d + e*x)^(1 + m)*AppellF1[1 + m, 3/2, 3/2 , 2 + m, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)]*e)])/ (e*(1 + m)*(a + c*x^2)^(3/2))
Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {514, 150}
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.
\(\displaystyle \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 514 |
\(\displaystyle \frac {\left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2} \int \frac {(d+e x)^m}{\left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2}}d(d+e x)}{e \left (a+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {(d+e x)^{m+1} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{3/2} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{3/2} \operatorname {AppellF1}\left (m+1,\frac {3}{2},\frac {3}{2},m+2,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (m+1) \left (a+c x^2\right )^{3/2}}\) |
((d + e*x)^(1 + m)*(1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^(3/2)*(1 - ( d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c]))^(3/2)*AppellF1[1 + m, 3/2, 3/2, 2 + m , (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt[-a]*e)/Sqrt[c ])])/(e*(1 + m)*(a + c*x^2)^(3/2))
3.8.30.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( c + d*x)/(c + d*q))^p) Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c^2 + a*d^2, 0]
\[\int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^m}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]