\(\int f^{a+b x^2} x^8 \, dx\) [84]

Optimal result

Integrand size = 13, antiderivative size = 128 \[ \int f^{a+b x^2} x^8 \, dx=\frac {105 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)} \]

[Out]

-105/16*f^(b*x^2+a)*x/b^4/ln(f)^4+35/8*f^(b*x^2+a)*x^3/b^3/ln(f)^3-7/4*f^(b*x^2+a)*x^5/b^2/ln(f)^2+1/2*f^(b*x^
2+a)*x^7/b/ln(f)+105/32*f^a*erfi(x*b^(1/2)*ln(f)^(1/2))*Pi^(1/2)/b^(9/2)/ln(f)^(9/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2243, 2235} \[ \int f^{a+b x^2} x^8 \, dx=\frac {105 \sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {105 x f^{a+b x^2}}{16 b^4 \log ^4(f)}+\frac {35 x^3 f^{a+b x^2}}{8 b^3 \log ^3(f)}-\frac {7 x^5 f^{a+b x^2}}{4 b^2 \log ^2(f)}+\frac {x^7 f^{a+b x^2}}{2 b \log (f)} \]

[In]

Int[f^(a + b*x^2)*x^8,x]

[Out]

(105*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]])/(32*b^(9/2)*Log[f]^(9/2)) - (105*f^(a + b*x^2)*x)/(16*b^4*Log[
f]^4) + (35*f^(a + b*x^2)*x^3)/(8*b^3*Log[f]^3) - (7*f^(a + b*x^2)*x^5)/(4*b^2*Log[f]^2) + (f^(a + b*x^2)*x^7)
/(2*b*Log[f])

Rule 2235

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]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x^2} x^7}{2 b \log (f)}-\frac {7 \int f^{a+b x^2} x^6 \, dx}{2 b \log (f)} \\ & = -\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}+\frac {35 \int f^{a+b x^2} x^4 \, dx}{4 b^2 \log ^2(f)} \\ & = \frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}-\frac {105 \int f^{a+b x^2} x^2 \, dx}{8 b^3 \log ^3(f)} \\ & = -\frac {105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)}+\frac {105 \int f^{a+b x^2} \, dx}{16 b^4 \log ^4(f)} \\ & = \frac {105 f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)}-\frac {105 f^{a+b x^2} x}{16 b^4 \log ^4(f)}+\frac {35 f^{a+b x^2} x^3}{8 b^3 \log ^3(f)}-\frac {7 f^{a+b x^2} x^5}{4 b^2 \log ^2(f)}+\frac {f^{a+b x^2} x^7}{2 b \log (f)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int f^{a+b x^2} x^8 \, dx=\frac {f^a \left (105 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )+2 \sqrt {b} f^{b x^2} x \sqrt {\log (f)} \left (-105+70 b x^2 \log (f)-28 b^2 x^4 \log ^2(f)+8 b^3 x^6 \log ^3(f)\right )\right )}{32 b^{9/2} \log ^{\frac {9}{2}}(f)} \]

[In]

Integrate[f^(a + b*x^2)*x^8,x]

[Out]

(f^a*(105*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]] + 2*Sqrt[b]*f^(b*x^2)*x*Sqrt[Log[f]]*(-105 + 70*b*x^2*Log[f] -
28*b^2*x^4*Log[f]^2 + 8*b^3*x^6*Log[f]^3)))/(32*b^(9/2)*Log[f]^(9/2))

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77

method	result	size
meijerg	\(\frac {f^{a} \left (-\frac {x \left (-b \right )^{\frac {9}{2}} \sqrt {\ln \left (f \right )}\, \left (-72 b^{3} x^{6} \ln \left (f \right )^{3}+252 b^{2} x^{4} \ln \left (f \right )^{2}-630 b \,x^{2} \ln \left (f \right )+945\right ) {\mathrm e}^{b \,x^{2} \ln \left (f \right )}}{72 b^{4}}+\frac {105 \left (-b \right )^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{16 b^{\frac {9}{2}}}\right )}{2 \ln \left (f \right )^{\frac {9}{2}} b^{4} \sqrt {-b}}\)	\(99\)
risch	\(\frac {f^{a} x^{7} f^{b \,x^{2}}}{2 \ln \left (f \right ) b}-\frac {7 f^{a} x^{5} f^{b \,x^{2}}}{4 \ln \left (f \right )^{2} b^{2}}+\frac {35 f^{a} x^{3} f^{b \,x^{2}}}{8 \ln \left (f \right )^{3} b^{3}}-\frac {105 f^{a} x \,f^{b \,x^{2}}}{16 \ln \left (f \right )^{4} b^{4}}+\frac {105 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x \right )}{32 \ln \left (f \right )^{4} b^{4} \sqrt {-b \ln \left (f \right )}}\)	\(120\)

[In]

int(f^(b*x^2+a)*x^8,x,method=_RETURNVERBOSE)

[Out]

1/2*f^a/ln(f)^(9/2)/b^4/(-b)^(1/2)*(-1/72*x*(-b)^(9/2)*ln(f)^(1/2)*(-72*b^3*x^6*ln(f)^3+252*b^2*x^4*ln(f)^2-63
0*b*x^2*ln(f)+945)/b^4*exp(b*x^2*ln(f))+105/16*(-b)^(9/2)/b^(9/2)*Pi^(1/2)*erfi(x*b^(1/2)*ln(f)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int f^{a+b x^2} x^8 \, dx=-\frac {105 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right ) - 2 \, {\left (8 \, b^{4} x^{7} \log \left (f\right )^{4} - 28 \, b^{3} x^{5} \log \left (f\right )^{3} + 70 \, b^{2} x^{3} \log \left (f\right )^{2} - 105 \, b x \log \left (f\right )\right )} f^{b x^{2} + a}}{32 \, b^{5} \log \left (f\right )^{5}} \]

[In]

integrate(f^(b*x^2+a)*x^8,x, algorithm="fricas")

[Out]

-1/32*(105*sqrt(pi)*sqrt(-b*log(f))*f^a*erf(sqrt(-b*log(f))*x) - 2*(8*b^4*x^7*log(f)^4 - 28*b^3*x^5*log(f)^3 +
70*b^2*x^3*log(f)^2 - 105*b*x*log(f))*f^(b*x^2 + a))/(b^5*log(f)^5)

Sympy [F]

\[ \int f^{a+b x^2} x^8 \, dx=\int f^{a + b x^{2}} x^{8}\, dx \]

[In]

integrate(f**(b*x**2+a)*x**8,x)

[Out]

Integral(f**(a + b*x**2)*x**8, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int f^{a+b x^2} x^8 \, dx=\frac {{\left (8 \, b^{3} f^{a} x^{7} \log \left (f\right )^{3} - 28 \, b^{2} f^{a} x^{5} \log \left (f\right )^{2} + 70 \, b f^{a} x^{3} \log \left (f\right ) - 105 \, f^{a} x\right )} f^{b x^{2}}}{16 \, b^{4} \log \left (f\right )^{4}} + \frac {105 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{32 \, \sqrt {-b \log \left (f\right )} b^{4} \log \left (f\right )^{4}} \]

[In]

integrate(f^(b*x^2+a)*x^8,x, algorithm="maxima")

[Out]

1/16*(8*b^3*f^a*x^7*log(f)^3 - 28*b^2*f^a*x^5*log(f)^2 + 70*b*f^a*x^3*log(f) - 105*f^a*x)*f^(b*x^2)/(b^4*log(f
)^4) + 105/32*sqrt(pi)*f^a*erf(sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^4*log(f)^4)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.72 \[ \int f^{a+b x^2} x^8 \, dx=-\frac {105 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} x\right )}{32 \, \sqrt {-b \log \left (f\right )} b^{4} \log \left (f\right )^{4}} + \frac {{\left (8 \, b^{3} x^{7} \log \left (f\right )^{3} - 28 \, b^{2} x^{5} \log \left (f\right )^{2} + 70 \, b x^{3} \log \left (f\right ) - 105 \, x\right )} e^{\left (b x^{2} \log \left (f\right ) + a \log \left (f\right )\right )}}{16 \, b^{4} \log \left (f\right )^{4}} \]

[In]

integrate(f^(b*x^2+a)*x^8,x, algorithm="giac")

[Out]

-105/32*sqrt(pi)*f^a*erf(-sqrt(-b*log(f))*x)/(sqrt(-b*log(f))*b^4*log(f)^4) + 1/16*(8*b^3*x^7*log(f)^3 - 28*b^
2*x^5*log(f)^2 + 70*b*x^3*log(f) - 105*x)*e^(b*x^2*log(f) + a*log(f))/(b^4*log(f)^4)

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int f^{a+b x^2} x^8 \, dx=\frac {\frac {f^a\,\left (105\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (f\right )}{\sqrt {b\,\ln \left (f\right )}}\right )-210\,f^{b\,x^2}\,x\,\sqrt {b\,\ln \left (f\right )}\right )}{32\,\sqrt {b\,\ln \left (f\right )}}-\frac {7\,b^2\,f^a\,f^{b\,x^2}\,x^5\,{\ln \left (f\right )}^2}{4}+\frac {b^3\,f^a\,f^{b\,x^2}\,x^7\,{\ln \left (f\right )}^3}{2}+\frac {35\,b\,f^a\,f^{b\,x^2}\,x^3\,\ln \left (f\right )}{8}}{b^4\,{\ln \left (f\right )}^4} \]

[In]

int(f^(a + b*x^2)*x^8,x)

[Out]

((f^a*(105*pi^(1/2)*erfi((b*x*log(f))/(b*log(f))^(1/2)) - 210*f^(b*x^2)*x*(b*log(f))^(1/2)))/(32*(b*log(f))^(1
/2)) - (7*b^2*f^a*f^(b*x^2)*x^5*log(f)^2)/4 + (b^3*f^a*f^(b*x^2)*x^7*log(f)^3)/2 + (35*b*f^a*f^(b*x^2)*x^3*log
(f))/8)/(b^4*log(f)^4)