Integrand size = 44, antiderivative size = 205 \[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (1+m)}+\frac {(b e g (1+2 m)-2 c (d g m+e f (1+m))) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (c d-b e-c e x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 e^2 (1+m) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
(b*e*g*(1+2*m)-2*c*(d*g*m+e*f*(1+m)))*(e*x+d)^m*(c*(e*x+d)/(-b*e+2*c*d))^( 1/2-m)*(-c*e*x-b*e+c*d)*hypergeom([1/2, 1/2-m],[3/2],(-c*e*x-b*e+c*d)/(-b* e+2*c*d))/c^2/e^2/(1+m)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-g*(e*x+d)^m *(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c/e^2/(1+m)
Time = 0.44 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (d+e x)^m \sqrt {(d+e x) (-b e+c (d-e x))} \left (e (e f-d g)+\frac {e (b e g (1+2 m)-2 c (d g m+e f (1+m))) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-\frac {1}{2}-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {3}{2},\frac {-c d+b e+c e x}{-2 c d+b e}\right )}{c}\right )}{e^3 (-2 c d+b e) (1+2 m)} \]
(-2*(d + e*x)^m*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(e*(e*f - d*g) + (e *(b*e*g*(1 + 2*m) - 2*c*(d*g*m + e*f*(1 + m)))*((c*(d + e*x))/(2*c*d - b*e ))^(-1/2 - m)*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (-(c*d) + b*e + c*e*x) /(-2*c*d + b*e)])/c))/(e^3*(-2*c*d + b*e)*(1 + 2*m))
Time = 0.48 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1221, 1139, 1138, 80, 79}
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.
\(\displaystyle \int \frac {(f+g x) (d+e x)^m}{\sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 1221 |
\(\displaystyle -\frac {(b e g (2 m+1)-2 c (d g m+e f (m+1))) \int \frac {(d+e x)^m}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c e (m+1)}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)}\) |
\(\Big \downarrow \) 1139 |
\(\displaystyle -\frac {(d+e x)^m \left (\frac {e x}{d}+1\right )^{-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \int \frac {\left (\frac {e x}{d}+1\right )^m}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c e (m+1)}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)}\) |
\(\Big \downarrow \) 1138 |
\(\displaystyle -\frac {(d+e x)^m \left (\frac {e x}{d}+1\right )^{\frac {1}{2}-m} \sqrt {d (c d-b e)-c d e x} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \int \frac {\left (\frac {e x}{d}+1\right )^{m-\frac {1}{2}}}{\sqrt {d (c d-b e)-c d e x}}dx}{2 c e (m+1) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle -\frac {(d+e x)^m \sqrt {d (c d-b e)-c d e x} \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \int \frac {\left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{m-\frac {1}{2}}}{\sqrt {d (c d-b e)-c d e x}}dx}{2 c e (m+1) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(d+e x)^m (d (c d-b e)-c d e x) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (2 m+1)-2 c (d g m+e f (m+1))) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {c d-b e-c e x}{2 c d-b e}\right )}{c^2 d e^2 (m+1) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g (d+e x)^m \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e^2 (m+1)}\) |
-((g*(d + e*x)^m*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c*e^2*(1 + m) )) + ((b*e*g*(1 + 2*m) - 2*c*(d*g*m + e*f*(1 + m)))*(d + e*x)^m*((c*(d + e *x))/(2*c*d - b*e))^(1/2 - m)*(d*(c*d - b*e) - c*d*e*x)*Hypergeometric2F1[ 1/2, 1/2 - m, 3/2, (c*d - b*e - c*e*x)/(2*c*d - b*e)])/(c^2*d*e^2*(1 + m)* Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
3.23.88.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] *(a/d + (c*x)/e)^FracPart[p])) Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[m] || GtQ[d, 0]) && !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer Q[4*p]))
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d^IntPart[m]*((d + e*x)^FracPart[m]/(1 + e*(x/d))^FracPart[m] ) Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !(IntegerQ[m] || GtQ[d, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
\[\int \frac {\left (e x +d \right )^{m} \left (g x +f \right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}d x\]
\[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}} \,d x } \]
integral(-sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)*(e*x + d)^m /(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e), x)
\[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{m} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]
\[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}} \,d x } \]
\[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^m (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^m}{\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \]