3.1.42 \(\int F^{c (a+b x)} \sqrt {d+e x} \, dx\) [42]

Optimal. Leaf size=105 \[ -\frac {\sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \]

[Out]

-1/2*F^(c*(a-b*d/e))*erfi(b^(1/2)*c^(1/2)*(e*x+d)^(1/2)*ln(F)^(1/2)/e^(1/2))*e^(1/2)*Pi^(1/2)/b^(3/2)/c^(3/2)/
ln(F)^(3/2)+F^(c*(b*x+a))*(e*x+d)^(1/2)/b/c/ln(F)

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Rubi [A]
time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2211, 2235} \begin {gather*} \frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Sqrt[d + e*x],x]

[Out]

-1/2*(Sqrt[e]*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(b^(3/2
)*c^(3/2)*Log[F]^(3/2)) + (F^(c*(a + b*x))*Sqrt[d + e*x])/(b*c*Log[F])

Rule 2207

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] - Dist[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]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

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]

Rubi steps

\begin {align*} \int F^{c (a+b x)} \sqrt {d+e x} \, dx &=\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {e \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{2 b c \log (F)}\\ &=\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {\text {Subst}\left (\int F^{c \left (a-\frac {b d}{e}\right )+\frac {b c x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b c \log (F)}\\ &=-\frac {\sqrt {e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 63, normalized size = 0.60 \begin {gather*} -\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{3/2} \Gamma \left (\frac {3}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*Sqrt[d + e*x],x]

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(3/2)*Gamma[3/2, -((b*c*(d + e*x)*Log[F])/e)])/(e*(-((b*c*(d + e*x)*Log[F])/e
))^(3/2)))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \sqrt {e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*(e*x+d)^(1/2),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)*F^((b*x + a)*c), x)

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Fricas [A]
time = 0.37, size = 91, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {x e + d} F^{b c x + a c} b c \log \left (F\right ) + \frac {\sqrt {\pi } \sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \operatorname {erf}\left (\sqrt {-b c e^{\left (-1\right )} \log \left (F\right )} \sqrt {x e + d}\right ) e}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(x*e + d)*F^(b*c*x + a*c)*b*c*log(F) + sqrt(pi)*sqrt(-b*c*e^(-1)*log(F))*erf(sqrt(-b*c*e^(-1)*log(F
))*sqrt(x*e + d))*e/F^((b*c*d - a*c*e)*e^(-1)))/(b^2*c^2*log(F)^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{c \left (a + b x\right )} \sqrt {d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(e*x+d)**(1/2),x)

[Out]

Integral(F**(c*(a + b*x))*sqrt(d + e*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (83) = 166\).
time = 4.24, size = 198, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, \sqrt {\pi } d \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )}} - \frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} \operatorname {erf}\left (-\sqrt {-b c e \log \left (F\right )} \sqrt {x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} - \frac {2 \, \sqrt {x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \left (F\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

-1/2*(2*sqrt(pi)*d*erf(-sqrt(-b*c*e*log(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*log(F) - a*c*e*log(F))*e^(-1) + 1
)/sqrt(-b*c*e*log(F)) - sqrt(pi)*(2*b*c*d*log(F) + e)*erf(-sqrt(-b*c*e*log(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*
d*log(F) - a*c*e*log(F))*e^(-1) + 1)/(sqrt(-b*c*e*log(F))*b*c*log(F)) - 2*sqrt(x*e + d)*e^(((x*e + d)*b*c*log(
F) - b*c*d*log(F) + a*c*e*log(F))*e^(-1) + 1)/(b*c*log(F)))*e^(-1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,\sqrt {d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*(d + e*x)^(1/2),x)

[Out]

int(F^(c*(a + b*x))*(d + e*x)^(1/2), x)

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