3.1.41 \(\int F^{c (a+b x)} (d+e x)^{3/2} \, dx\) [41]

Optimal. Leaf size=138 \[ \frac {3 e^{3/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)} \]

[Out]

F^(c*(b*x+a))*(e*x+d)^(3/2)/b/c/ln(F)+3/4*e^(3/2)*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))*Pi^(1/2)/b^(5/2)/c^(5/2)/ln(F)^(5/2)-3/2*e*F^(c*(b*x+a))*(e*x+d)^(1/2)/b^2/c^2/ln(F)^2

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {3 \sqrt {\pi } e^{3/2} 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 )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e \sqrt {d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac {(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e^(3/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[Log[F]])/Sqrt[e]])/(4*b^(5/2)
*c^(5/2)*Log[F]^(5/2)) - (3*e*F^(c*(a + b*x))*Sqrt[d + e*x])/(2*b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)
^(3/2))/(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)} (d+e x)^{3/2} \, dx &=\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}-\frac {(3 e) \int F^{c (a+b x)} \sqrt {d+e x} \, dx}{2 b c \log (F)}\\ &=-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac {\left (3 e^2\right ) \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{4 b^2 c^2 \log ^2(F)}\\ &=-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac {(3 e) \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 )}{2 b^2 c^2 \log ^2(F)}\\ &=\frac {3 e^{3/2} 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 )}{4 b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F)}-\frac {3 e F^{c (a+b x)} \sqrt {d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 63, normalized size = 0.46 \begin {gather*} -\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{5/2} \Gamma \left (\frac {5}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((F^(c*(a - (b*d)/e))*(d + e*x)^(5/2)*Gamma[5/2, -((b*c*(d + e*x)*Log[F])/e)])/(e*(-((b*c*(d + e*x)*Log[F])/e
))^(5/2)))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{c \left (b x +a \right )} \left (e x +d \right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*(e*x+d)^(3/2),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(3/2),x)

________________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((x*e + d)^(3/2)*F^((b*x + a)*c), x)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 122, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (3 \, b c e \log \left (F\right ) - 2 \, {\left (b^{2} c^{2} x e + b^{2} c^{2} d\right )} \log \left (F\right )^{2}\right )} \sqrt {x e + d} F^{b c x + a c} + \frac {3 \, \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^{2}}{F^{{\left (b c d - a c e\right )} e^{\left (-1\right )}}}}{4 \, b^{3} c^{3} \log \left (F\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(2*(3*b*c*e*log(F) - 2*(b^2*c^2*x*e + b^2*c^2*d)*log(F)^2)*sqrt(x*e + d)*F^(b*c*x + a*c) + 3*sqrt(pi)*sqr
t(-b*c*e^(-1)*log(F))*erf(sqrt(-b*c*e^(-1)*log(F))*sqrt(x*e + d))*e^2/F^((b*c*d - a*c*e)*e^(-1)))/(b^3*c^3*log
(F)^3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*(e*x+d)**(3/2),x)

[Out]

Integral(F**(c*(a + b*x))*(d + e*x)**(3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (114) = 228\).
time = 3.53, size = 401, normalized size = 2.91 \begin {gather*} -\frac {1}{4} \, {\left (\frac {4 \, \sqrt {\pi } d^{2} \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 )}} - 4 \, d {\left (\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 )} + {\left (\frac {\sqrt {\pi } {\left (4 \, b^{2} c^{2} d^{2} \log \left (F\right )^{2} + 4 \, b c d e \log \left (F\right ) + 3 \, e^{2}\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 ) + 2 \, e\right )} e^{\left (-1\right )} + 1\right )}}{\sqrt {-b c e \log \left (F\right )} b^{2} c^{2} \log \left (F\right )^{2}} - \frac {2 \, {\left (2 \, {\left (x e + d\right )}^{\frac {3}{2}} b c e \log \left (F\right ) - 4 \, \sqrt {x e + d} b c d e \log \left (F\right ) - 3 \, \sqrt {x e + d} e^{2}\right )} 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 ) - 2 \, e\right )} e^{\left (-1\right )}\right )}}{b^{2} c^{2} \log \left (F\right )^{2}}\right )} e^{2}\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)^(3/2),x, algorithm="giac")

[Out]

-1/4*(4*sqrt(pi)*d^2*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)) - 4*d*(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))) + (sqrt(pi)*(4*b^2*c^2*d^2*log(F)^2 + 4*b*c
*d*e*log(F) + 3*e^2)*erf(-sqrt(-b*c*e*log(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*log(F) - a*c*e*log(F) + 2*e)*e^
(-1) + 1)/(sqrt(-b*c*e*log(F))*b^2*c^2*log(F)^2) - 2*(2*(x*e + d)^(3/2)*b*c*e*log(F) - 4*sqrt(x*e + d)*b*c*d*e
*log(F) - 3*sqrt(x*e + d)*e^2)*e^(((x*e + d)*b*c*log(F) - b*c*d*log(F) + a*c*e*log(F) - 2*e)*e^(-1))/(b^2*c^2*
log(F)^2))*e^2)*e^(-1)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))*(d + e*x)^(3/2),x)

[Out]

int(F^(c*(a + b*x))*(d + e*x)^(3/2), x)

________________________________________________________________________________________