Integrand size = 13, antiderivative size = 131 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {105 F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{16 b^{9/2} \log ^{\frac {9}{2}}(F)}-\frac {105 F^{a+b x} \sqrt {x}}{8 b^4 \log ^4(F)}+\frac {35 F^{a+b x} x^{3/2}}{4 b^3 \log ^3(F)}-\frac {7 F^{a+b x} x^{5/2}}{2 b^2 \log ^2(F)}+\frac {F^{a+b x} x^{7/2}}{b \log (F)} \] Output:
105/16*F^a*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2))/b^(9/2)/ln(F)^(9/2)- 105/8*F^(b*x+a)*x^(1/2)/b^4/ln(F)^4+35/4*F^(b*x+a)*x^(3/2)/b^3/ln(F)^3-7/2 *F^(b*x+a)*x^(5/2)/b^2/ln(F)^2+F^(b*x+a)*x^(7/2)/b/ln(F)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.27 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {F^a \Gamma \left (\frac {9}{2},-b x \log (F)\right ) \sqrt {-b x \log (F)}}{b^5 \sqrt {x} \log ^5(F)} \] Input:
Integrate[F^(a + b*x)*x^(7/2),x]
Output:
(F^a*Gamma[9/2, -(b*x*Log[F])]*Sqrt[-(b*x*Log[F])])/(b^5*Sqrt[x]*Log[F]^5)
Time = 0.77 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2607, 2607, 2607, 2607, 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.
\(\displaystyle \int x^{7/2} F^{a+b x} \, dx\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \int F^{a+b x} x^{5/2}dx}{2 b \log (F)}\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \left (\frac {x^{5/2} F^{a+b x}}{b \log (F)}-\frac {5 \int F^{a+b x} x^{3/2}dx}{2 b \log (F)}\right )}{2 b \log (F)}\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \left (\frac {x^{5/2} F^{a+b x}}{b \log (F)}-\frac {5 \left (\frac {x^{3/2} F^{a+b x}}{b \log (F)}-\frac {3 \int F^{a+b x} \sqrt {x}dx}{2 b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \left (\frac {x^{5/2} F^{a+b x}}{b \log (F)}-\frac {5 \left (\frac {x^{3/2} F^{a+b x}}{b \log (F)}-\frac {3 \left (\frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\int \frac {F^{a+b x}}{\sqrt {x}}dx}{2 b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \left (\frac {x^{5/2} F^{a+b x}}{b \log (F)}-\frac {5 \left (\frac {x^{3/2} F^{a+b x}}{b \log (F)}-\frac {3 \left (\frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\int F^{a+b x}d\sqrt {x}}{b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {x^{7/2} F^{a+b x}}{b \log (F)}-\frac {7 \left (\frac {x^{5/2} F^{a+b x}}{b \log (F)}-\frac {5 \left (\frac {x^{3/2} F^{a+b x}}{b \log (F)}-\frac {3 \left (\frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\right )}{2 b \log (F)}\) |
Input:
Int[F^(a + b*x)*x^(7/2),x]
Output:
(F^(a + b*x)*x^(7/2))/(b*Log[F]) - (7*((F^(a + b*x)*x^(5/2))/(b*Log[F]) - (5*((F^(a + b*x)*x^(3/2))/(b*Log[F]) - (3*(-1/2*(F^a*Sqrt[Pi]*Erfi[Sqrt[b] *Sqrt[x]*Sqrt[Log[F]]])/(b^(3/2)*Log[F]^(3/2)) + (F^(a + b*x)*Sqrt[x])/(b* Log[F])))/(2*b*Log[F])))/(2*b*Log[F])))/(2*b*Log[F])
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Simp[d*(m/(f*g*n*Log[F])) 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] && GtQ[m, 0] && IntegerQ[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]
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76
method | result | size |
meijerg | \(-\frac {F^{a} \left (-\frac {\sqrt {x}\, \left (-b \right )^{\frac {9}{2}} \sqrt {\ln \left (F \right )}\, \left (-72 b^{3} x^{3} \ln \left (F \right )^{3}+252 b^{2} x^{2} \ln \left (F \right )^{2}-630 \ln \left (F \right ) b x +945\right ) {\mathrm e}^{\ln \left (F \right ) b x}}{72 b^{4}}+\frac {105 \left (-b \right )^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{16 b^{\frac {9}{2}}}\right )}{\left (-b \right )^{\frac {7}{2}} \ln \left (F \right )^{\frac {9}{2}} b}\) | \(99\) |
Input:
int(F^(b*x+a)*x^(7/2),x,method=_RETURNVERBOSE)
Output:
-F^a/(-b)^(7/2)/ln(F)^(9/2)/b*(-1/72*x^(1/2)*(-b)^(9/2)*ln(F)^(1/2)*(-72*b ^3*x^3*ln(F)^3+252*b^2*x^2*ln(F)^2-630*ln(F)*b*x+945)/b^4*exp(ln(F)*b*x)+1 05/16*(-b)^(9/2)/b^(9/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.68 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {105 \, \sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right ) - 2 \, {\left (8 \, b^{4} x^{3} \log \left (F\right )^{4} - 28 \, b^{3} x^{2} \log \left (F\right )^{3} + 70 \, b^{2} x \log \left (F\right )^{2} - 105 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt {x}}{16 \, b^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^(b*x+a)*x^(7/2),x, algorithm="fricas")
Output:
-1/16*(105*sqrt(pi)*sqrt(-b*log(F))*F^a*erf(sqrt(-b*log(F))*sqrt(x)) - 2*( 8*b^4*x^3*log(F)^4 - 28*b^3*x^2*log(F)^3 + 70*b^2*x*log(F)^2 - 105*b*log(F ))*F^(b*x + a)*sqrt(x))/(b^5*log(F)^5)
\[ \int F^{a+b x} x^{7/2} \, dx=\int F^{a + b x} x^{\frac {7}{2}}\, dx \] Input:
integrate(F**(b*x+a)*x**(7/2),x)
Output:
Integral(F**(a + b*x)*x**(7/2), x)
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.18 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {F^{a} x^{\frac {9}{2}} \Gamma \left (\frac {9}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac {9}{2}}} \] Input:
integrate(F^(b*x+a)*x^(7/2),x, algorithm="maxima")
Output:
-F^a*x^(9/2)*gamma(9/2, -b*x*log(F))/(-b*x*log(F))^(9/2)
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int F^{a+b x} x^{7/2} \, dx=-\frac {105 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{16 \, \sqrt {-b \log \left (F\right )} b^{4} \log \left (F\right )^{4}} + \frac {{\left (8 \, b^{3} x^{\frac {7}{2}} \log \left (F\right )^{3} - 28 \, b^{2} x^{\frac {5}{2}} \log \left (F\right )^{2} + 70 \, b x^{\frac {3}{2}} \log \left (F\right ) - 105 \, \sqrt {x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{8 \, b^{4} \log \left (F\right )^{4}} \] Input:
integrate(F^(b*x+a)*x^(7/2),x, algorithm="giac")
Output:
-105/16*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*sqrt(x))/(sqrt(-b*log(F))*b^4*lo g(F)^4) + 1/8*(8*b^3*x^(7/2)*log(F)^3 - 28*b^2*x^(5/2)*log(F)^2 + 70*b*x^( 3/2)*log(F) - 105*sqrt(x))*e^(b*x*log(F) + a*log(F))/(b^4*log(F)^4)
Time = 23.42 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.63 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {F^a\,x^{7/2}\,\left (\frac {105\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )}{16}+F^{b\,x}\,\left (\frac {105\,\sqrt {-b\,x\,\ln \left (F\right )}}{8}+\frac {35\,{\left (-b\,x\,\ln \left (F\right )\right )}^{3/2}}{4}+\frac {7\,{\left (-b\,x\,\ln \left (F\right )\right )}^{5/2}}{2}+{\left (-b\,x\,\ln \left (F\right )\right )}^{7/2}\right )\right )}{b\,\ln \left (F\right )\,{\left (-b\,x\,\ln \left (F\right )\right )}^{7/2}} \] Input:
int(F^(a + b*x)*x^(7/2),x)
Output:
(F^a*x^(7/2)*((105*pi^(1/2)*erfc((-b*x*log(F))^(1/2)))/16 + F^(b*x)*((105* (-b*x*log(F))^(1/2))/8 + (35*(-b*x*log(F))^(3/2))/4 + (7*(-b*x*log(F))^(5/ 2))/2 + (-b*x*log(F))^(7/2))))/(b*log(F)*(-b*x*log(F))^(7/2))
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int F^{a+b x} x^{7/2} \, dx=\frac {f^{a} \left (-105 \sqrt {\pi }\, \mathrm {erf}\left (\sqrt {x}\, \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\, i \right ) i +16 \sqrt {x}\, f^{b x} \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\, \mathrm {log}\left (f \right )^{3} b^{3} x^{3}-56 \sqrt {x}\, f^{b x} \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\, \mathrm {log}\left (f \right )^{2} b^{2} x^{2}+140 \sqrt {x}\, f^{b x} \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\, \mathrm {log}\left (f \right ) b x -210 \sqrt {x}\, f^{b x} \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\right )}{16 \sqrt {b}\, \sqrt {\mathrm {log}\left (f \right )}\, \mathrm {log}\left (f \right )^{4} b^{4}} \] Input:
int(F^(b*x+a)*x^(7/2),x)
Output:
(f**a*( - 105*sqrt(pi)*erf(sqrt(x)*sqrt(b)*sqrt(log(f))*i)*i + 16*sqrt(x)* f**(b*x)*sqrt(b)*sqrt(log(f))*log(f)**3*b**3*x**3 - 56*sqrt(x)*f**(b*x)*sq rt(b)*sqrt(log(f))*log(f)**2*b**2*x**2 + 140*sqrt(x)*f**(b*x)*sqrt(b)*sqrt (log(f))*log(f)*b*x - 210*sqrt(x)*f**(b*x)*sqrt(b)*sqrt(log(f))))/(16*sqrt (b)*sqrt(log(f))*log(f)**4*b**4)