Integrand size = 19, antiderivative size = 97 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}+\frac {2 \sqrt {b} \sqrt {c} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right ) \sqrt {\log (F)}}{e^{3/2}} \] Output:
-2*F^(c*(b*x+a))/e/(e*x+d)^(1/2)+2*b^(1/2)*c^(1/2)*F^(c*(a-b*d/e))*Pi^(1/2 )*erfi(b^(1/2)*c^(1/2)*(e*x+d)^(1/2)*ln(F)^(1/2)/e^(1/2))*ln(F)^(1/2)/e^(3 /2)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (F^{c (a+b x)}-F^{c \left (a-\frac {b d}{e}\right )} \Gamma \left (\frac {1}{2},-\frac {b c (d+e x) \log (F)}{e}\right ) \sqrt {-\frac {b c (d+e x) \log (F)}{e}}\right )}{e \sqrt {d+e x}} \] Input:
Integrate[F^(c*(a + b*x))/(d + e*x)^(3/2),x]
Output:
(-2*(F^(c*(a + b*x)) - F^(c*(a - (b*d)/e))*Gamma[1/2, -((b*c*(d + e*x)*Log [F])/e)]*Sqrt[-((b*c*(d + e*x)*Log[F])/e)]))/(e*Sqrt[d + e*x])
Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2608, 2611, 2633}
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.
\(\displaystyle \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle \frac {2 b c \log (F) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}}dx}{e}-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {4 b c \log (F) \int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c (d+e x)}{e}}d\sqrt {d+e x}}{e^2}-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2 \sqrt {\pi } \sqrt {b} \sqrt {c} \sqrt {\log (F)} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{e^{3/2}}-\frac {2 F^{c (a+b x)}}{e \sqrt {d+e x}}\) |
Input:
Int[F^(c*(a + b*x))/(d + e*x)^(3/2),x]
Output:
(-2*F^(c*(a + b*x)))/(e*Sqrt[d + e*x]) + (2*Sqrt[b]*Sqrt[c]*F^(c*(a - (b*d )/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]]* Sqrt[Log[F]])/e^(3/2)
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))) , x] - Simp[f*g*n*(Log[F]/(d*(m + 1))) Int[(c + d*x)^(m + 1)*(b*F^(g*(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && In tegerQ[2*m] && !TrueQ[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
\[\int \frac {F^{c \left (b x +a \right )}}{\left (e x +d \right )^{\frac {3}{2}}}d x\]
Input:
int(F^(c*(b*x+a))/(e*x+d)^(3/2),x)
Output:
int(F^(c*(b*x+a))/(e*x+d)^(3/2),x)
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {\sqrt {\pi } {\left (e x + d\right )} \sqrt {-\frac {b c \log \left (F\right )}{e}} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}} + \sqrt {e x + d} F^{b c x + a c}\right )}}{e^{2} x + d e} \] Input:
integrate(F^((b*x+a)*c)/(e*x+d)^(3/2),x, algorithm="fricas")
Output:
-2*(sqrt(pi)*(e*x + d)*sqrt(-b*c*log(F)/e)*erf(sqrt(e*x + d)*sqrt(-b*c*log (F)/e))/F^((b*c*d - a*c*e)/e) + sqrt(e*x + d)*F^(b*c*x + a*c))/(e^2*x + d* e)
\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=\int \frac {F^{c \left (a + b x\right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(F**((b*x+a)*c)/(e*x+d)**(3/2),x)
Output:
Integral(F**(c*(a + b*x))/(d + e*x)**(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(F^((b*x+a)*c)/(e*x+d)^(3/2),x, algorithm="maxima")
Output:
integrate(F^((b*x + a)*c)/(e*x + d)^(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(F^((b*x+a)*c)/(e*x+d)^(3/2),x, algorithm="giac")
Output:
integrate(F^((b*x + a)*c)/(e*x + d)^(3/2), x)
Timed out. \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int(F^(c*(a + b*x))/(d + e*x)^(3/2),x)
Output:
int(F^(c*(a + b*x))/(d + e*x)^(3/2), x)
\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx=f^{a c} \left (\int \frac {f^{b c x}}{\sqrt {e x +d}\, d +\sqrt {e x +d}\, e x}d x \right ) \] Input:
int(F^((b*x+a)*c)/(e*x+d)^(3/2),x)
Output:
f**(a*c)*int(f**(b*c*x)/(sqrt(d + e*x)*d + sqrt(d + e*x)*e*x),x)