3.40 \(\int F^{c (a+b x)} (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{15 \sqrt{\pi } e^{5/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{8 b^{7/2} c^{7/2} \log ^{\frac{7}{2}}(F)}+\frac{15 e^2 \sqrt{d+e x} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{5 e (d+e x)^{3/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(-15*e^(5/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sq
rt[Log[F]])/Sqrt[e]])/(8*b^(7/2)*c^(7/2)*Log[F]^(7/2)) + (15*e^2*F^(c*(a + b*x))
*Sqrt[d + e*x])/(4*b^3*c^3*Log[F]^3) - (5*e*F^(c*(a + b*x))*(d + e*x)^(3/2))/(2*
b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(5/2))/(b*c*Log[F])

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Rubi [A]  time = 0.262062, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{15 \sqrt{\pi } e^{5/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{8 b^{7/2} c^{7/2} \log ^{\frac{7}{2}}(F)}+\frac{15 e^2 \sqrt{d+e x} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{5 e (d+e x)^{3/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{5/2} F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x)^(5/2),x]

[Out]

(-15*e^(5/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sq
rt[Log[F]])/Sqrt[e]])/(8*b^(7/2)*c^(7/2)*Log[F]^(7/2)) + (15*e^2*F^(c*(a + b*x))
*Sqrt[d + e*x])/(4*b^3*c^3*Log[F]^3) - (5*e*F^(c*(a + b*x))*(d + e*x)^(3/2))/(2*
b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^(5/2))/(b*c*Log[F])

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Rubi in Sympy [A]  time = 45.0367, size = 167, normalized size = 0.97 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac{5}{2}}}{b c \log{\left (F \right )}} - \frac{5 F^{c \left (a + b x\right )} e \left (d + e x\right )^{\frac{3}{2}}}{2 b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{15 F^{c \left (a + b x\right )} e^{2} \sqrt{d + e x}}{4 b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{15 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} e^{\frac{5}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{8 b^{\frac{7}{2}} c^{\frac{7}{2}} \log{\left (F \right )}^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**(5/2),x)

[Out]

F**(c*(a + b*x))*(d + e*x)**(5/2)/(b*c*log(F)) - 5*F**(c*(a + b*x))*e*(d + e*x)*
*(3/2)/(2*b**2*c**2*log(F)**2) + 15*F**(c*(a + b*x))*e**2*sqrt(d + e*x)/(4*b**3*
c**3*log(F)**3) - 15*sqrt(pi)*F**(c*(a*e - b*d)/e)*e**(5/2)*erfi(sqrt(b)*sqrt(c)
*sqrt(d + e*x)*sqrt(log(F))/sqrt(e))/(8*b**(7/2)*c**(7/2)*log(F)**(7/2))

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Mathematica [A]  time = 0.438615, size = 204, normalized size = 1.18 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \left (8 b^3 c^3 \log ^3(F) (d+e x)^3 F^{\frac{b c (d+e x)}{e}}-20 b^2 c^2 e \log ^2(F) (d+e x)^2 F^{\frac{b c (d+e x)}{e}}+15 \sqrt{\pi } e^3 \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )-15 \sqrt{\pi } e^3 \sqrt{-\frac{b c \log (F) (d+e x)}{e}}+30 b c e^2 \log (F) (d+e x) F^{\frac{b c (d+e x)}{e}}\right )}{8 b^4 c^4 \log ^4(F) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x)^(5/2),x]

[Out]

(F^(c*(a - (b*d)/e))*(30*b*c*e^2*F^((b*c*(d + e*x))/e)*(d + e*x)*Log[F] - 20*b^2
*c^2*e*F^((b*c*(d + e*x))/e)*(d + e*x)^2*Log[F]^2 + 8*b^3*c^3*F^((b*c*(d + e*x))
/e)*(d + e*x)^3*Log[F]^3 - 15*e^3*Sqrt[Pi]*Sqrt[-((b*c*(d + e*x)*Log[F])/e)] + 1
5*e^3*Sqrt[Pi]*Erf[Sqrt[-((b*c*(d + e*x)*Log[F])/e)]]*Sqrt[-((b*c*(d + e*x)*Log[
F])/e)]))/(8*b^4*c^4*Sqrt[d + e*x]*Log[F]^4)

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^(5/2),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(5/2),x)

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Maxima [A]  time = 0.834722, size = 211, normalized size = 1.22 \[ -\frac{F^{a c}{\left (\frac{15 \, \sqrt{\pi } e^{3} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} F^{\frac{b c d}{e}} b^{3} c^{3} \log \left (F\right )^{3}} - \frac{2 \,{\left (4 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{2} c^{2} e \log \left (F\right )^{2} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} b c e^{2} \log \left (F\right ) + 15 \, \sqrt{e x + d} e^{3}\right )} F^{\frac{{\left (e x + d\right )} b c}{e}}}{F^{\frac{b c d}{e}} b^{3} c^{3} \log \left (F\right )^{3}}\right )}}{8 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

-1/8*F^(a*c)*(15*sqrt(pi)*e^3*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/(sqrt(-b*c*
log(F)/e)*F^(b*c*d/e)*b^3*c^3*log(F)^3) - 2*(4*(e*x + d)^(5/2)*b^2*c^2*e*log(F)^
2 - 10*(e*x + d)^(3/2)*b*c*e^2*log(F) + 15*sqrt(e*x + d)*e^3)*F^((e*x + d)*b*c/e
)/(F^(b*c*d/e)*b^3*c^3*log(F)^3))/e

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Fricas [A]  time = 0.268556, size = 221, normalized size = 1.28 \[ \frac{2 \,{\left (4 \,{\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + 15 \, e^{2} - 10 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} \sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c} - \frac{15 \, \sqrt{\pi } e^{2} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}}}}{8 \, \sqrt{-\frac{b c \log \left (F\right )}{e}} b^{3} c^{3} \log \left (F\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

1/8*(2*(4*(b^2*c^2*e^2*x^2 + 2*b^2*c^2*d*e*x + b^2*c^2*d^2)*log(F)^2 + 15*e^2 -
10*(b*c*e^2*x + b*c*d*e)*log(F))*sqrt(e*x + d)*sqrt(-b*c*log(F)/e)*F^(b*c*x + a*
c) - 15*sqrt(pi)*e^2*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/F^((b*c*d - a*c*e)/e
))/(sqrt(-b*c*log(F)/e)*b^3*c^3*log(F)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))*(e*x+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.264259, size = 779, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)*F^((b*x + a)*c),x, algorithm="giac")

[Out]

1/8*(4*d^2*(sqrt(pi)*erf(-sqrt(-b*c*e*ln(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*ln
(F) - a*c*e*ln(F))*e^(-1) + 2)/(sqrt(-b*c*e*ln(F))*b*c*ln(F)) + 2*sqrt(x*e + d)*
e^(((x*e + d)*b*c*ln(F) - b*c*d*ln(F) + a*c*e*ln(F))*e^(-1) + 1)/(b*c*ln(F))) -
4*d*(sqrt(pi)*(2*b*c*d*e*ln(F) + 3*e^2)*erf(-sqrt(-b*c*e*ln(F))*sqrt(x*e + d)*e^
(-1))*e^(-(b*c*d*ln(F) - a*c*e*ln(F))*e^(-1) + 1)/(sqrt(-b*c*e*ln(F))*b^2*c^2*ln
(F)^2) - 2*(2*(x*e + d)^(3/2)*b*c*e*ln(F) - 2*sqrt(x*e + d)*b*c*d*e*ln(F) - 3*sq
rt(x*e + d)*e^2)*e^(((x*e + d)*b*c*ln(F) - b*c*d*ln(F) + a*c*e*ln(F))*e^(-1))/(b
^2*c^2*ln(F)^2)) + (sqrt(pi)*(4*b^2*c^2*d^2*e*ln(F)^2 + 12*b*c*d*e^2*ln(F) + 15*
e^3)*erf(-sqrt(-b*c*e*ln(F))*sqrt(x*e + d)*e^(-1))*e^(-(b*c*d*ln(F) - a*c*e*ln(F
) + 2*e)*e^(-1) + 1)/(sqrt(-b*c*e*ln(F))*b^3*c^3*ln(F)^3) + 2*(4*(x*e + d)^(5/2)
*b^2*c^2*e*ln(F)^2 - 8*(x*e + d)^(3/2)*b^2*c^2*d*e*ln(F)^2 + 4*sqrt(x*e + d)*b^2
*c^2*d^2*e*ln(F)^2 - 10*(x*e + d)^(3/2)*b*c*e^2*ln(F) + 12*sqrt(x*e + d)*b*c*d*e
^2*ln(F) + 15*sqrt(x*e + d)*e^3)*e^(((x*e + d)*b*c*ln(F) - b*c*d*ln(F) + a*c*e*l
n(F) - 2*e)*e^(-1))/(b^3*c^3*ln(F)^3))*e^2)*e^(-1)