3.39 \(\int F^{c (a+b x)} (d+e x)^{7/2} \, dx\)

Optimal. Leaf size=208 \[ \frac{105 \sqrt{\pi } e^{7/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 e^3 \sqrt{d+e x} F^{c (a+b x)}}{8 b^4 c^4 \log ^4(F)}+\frac{35 e^2 (d+e x)^{3/2} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e (d+e x)^{5/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(105*e^(7/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sq
rt[Log[F]])/Sqrt[e]])/(16*b^(9/2)*c^(9/2)*Log[F]^(9/2)) - (105*e^3*F^(c*(a + b*x
))*Sqrt[d + e*x])/(8*b^4*c^4*Log[F]^4) + (35*e^2*F^(c*(a + b*x))*(d + e*x)^(3/2)
)/(4*b^3*c^3*Log[F]^3) - (7*e*F^(c*(a + b*x))*(d + e*x)^(5/2))/(2*b^2*c^2*Log[F]
^2) + (F^(c*(a + b*x))*(d + e*x)^(7/2))/(b*c*Log[F])

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Rubi [A]  time = 0.445157, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{105 \sqrt{\pi } e^{7/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 e^3 \sqrt{d+e x} F^{c (a+b x)}}{8 b^4 c^4 \log ^4(F)}+\frac{35 e^2 (d+e x)^{3/2} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e (d+e x)^{5/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x)^(7/2),x]

[Out]

(105*e^(7/2)*F^(c*(a - (b*d)/e))*Sqrt[Pi]*Erfi[(Sqrt[b]*Sqrt[c]*Sqrt[d + e*x]*Sq
rt[Log[F]])/Sqrt[e]])/(16*b^(9/2)*c^(9/2)*Log[F]^(9/2)) - (105*e^3*F^(c*(a + b*x
))*Sqrt[d + e*x])/(8*b^4*c^4*Log[F]^4) + (35*e^2*F^(c*(a + b*x))*(d + e*x)^(3/2)
)/(4*b^3*c^3*Log[F]^3) - (7*e*F^(c*(a + b*x))*(d + e*x)^(5/2))/(2*b^2*c^2*Log[F]
^2) + (F^(c*(a + b*x))*(d + e*x)^(7/2))/(b*c*Log[F])

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Rubi in Sympy [A]  time = 61.8135, size = 202, normalized size = 0.97 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{\frac{7}{2}}}{b c \log{\left (F \right )}} - \frac{7 F^{c \left (a + b x\right )} e \left (d + e x\right )^{\frac{5}{2}}}{2 b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{35 F^{c \left (a + b x\right )} e^{2} \left (d + e x\right )^{\frac{3}{2}}}{4 b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{105 F^{c \left (a + b x\right )} e^{3} \sqrt{d + e x}}{8 b^{4} c^{4} \log{\left (F \right )}^{4}} + \frac{105 \sqrt{\pi } F^{\frac{c \left (a e - b d\right )}{e}} e^{\frac{7}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d + e x} \sqrt{\log{\left (F \right )}}}{\sqrt{e}} \right )}}{16 b^{\frac{9}{2}} c^{\frac{9}{2}} \log{\left (F \right )}^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**(7/2),x)

[Out]

F**(c*(a + b*x))*(d + e*x)**(7/2)/(b*c*log(F)) - 7*F**(c*(a + b*x))*e*(d + e*x)*
*(5/2)/(2*b**2*c**2*log(F)**2) + 35*F**(c*(a + b*x))*e**2*(d + e*x)**(3/2)/(4*b*
*3*c**3*log(F)**3) - 105*F**(c*(a + b*x))*e**3*sqrt(d + e*x)/(8*b**4*c**4*log(F)
**4) + 105*sqrt(pi)*F**(c*(a*e - b*d)/e)*e**(7/2)*erfi(sqrt(b)*sqrt(c)*sqrt(d +
e*x)*sqrt(log(F))/sqrt(e))/(16*b**(9/2)*c**(9/2)*log(F)**(9/2))

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Mathematica [A]  time = 0.779436, size = 239, normalized size = 1.15 \[ \frac{F^{c \left (a-\frac{b d}{e}\right )} \left (16 b^4 c^4 \log ^4(F) (d+e x)^4 F^{\frac{b c (d+e x)}{e}}-56 b^3 c^3 e \log ^3(F) (d+e x)^3 F^{\frac{b c (d+e x)}{e}}+140 b^2 c^2 e^2 \log ^2(F) (d+e x)^2 F^{\frac{b c (d+e x)}{e}}-105 \sqrt{\pi } e^4 \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \text{Erf}\left (\sqrt{-\frac{b c \log (F) (d+e x)}{e}}\right )+105 \sqrt{\pi } e^4 \sqrt{-\frac{b c \log (F) (d+e x)}{e}}-210 b c e^3 \log (F) (d+e x) F^{\frac{b c (d+e x)}{e}}\right )}{16 b^5 c^5 \log ^5(F) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x)^(7/2),x]

[Out]

(F^(c*(a - (b*d)/e))*(-210*b*c*e^3*F^((b*c*(d + e*x))/e)*(d + e*x)*Log[F] + 140*
b^2*c^2*e^2*F^((b*c*(d + e*x))/e)*(d + e*x)^2*Log[F]^2 - 56*b^3*c^3*e*F^((b*c*(d
+ e*x))/e)*(d + e*x)^3*Log[F]^3 + 16*b^4*c^4*F^((b*c*(d + e*x))/e)*(d + e*x)^4*
Log[F]^4 + 105*e^4*Sqrt[Pi]*Sqrt[-((b*c*(d + e*x)*Log[F])/e)] - 105*e^4*Sqrt[Pi]
*Erf[Sqrt[-((b*c*(d + e*x)*Log[F])/e)]]*Sqrt[-((b*c*(d + e*x)*Log[F])/e)]))/(16*
b^5*c^5*Sqrt[d + e*x]*Log[F]^5)

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^(7/2),x)

[Out]

int(F^(c*(b*x+a))*(e*x+d)^(7/2),x)

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Maxima [A]  time = 0.821356, size = 240, normalized size = 1.15 \[ \frac{F^{a c}{\left (\frac{105 \, \sqrt{\pi } e^{4} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{\sqrt{-\frac{b c \log \left (F\right )}{e}} F^{\frac{b c d}{e}} b^{4} c^{4} \log \left (F\right )^{4}} + \frac{2 \,{\left (8 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{3} c^{3} e \log \left (F\right )^{3} - 28 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{2} c^{2} e^{2} \log \left (F\right )^{2} + 70 \,{\left (e x + d\right )}^{\frac{3}{2}} b c e^{3} \log \left (F\right ) - 105 \, \sqrt{e x + d} e^{4}\right )} F^{\frac{{\left (e x + d\right )} b c}{e}}}{F^{\frac{b c d}{e}} b^{4} c^{4} \log \left (F\right )^{4}}\right )}}{16 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.30967, size = 306, normalized size = 1.47 \[ \frac{\frac{105 \, \sqrt{\pi } e^{3} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}}} + 2 \,{\left (8 \,{\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )} \log \left (F\right )^{3} - 105 \, e^{3} - 28 \,{\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} + 70 \,{\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} \sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}} F^{b c x + a c}}{16 \, \sqrt{-\frac{b c \log \left (F\right )}{e}} b^{4} c^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

1/16*(105*sqrt(pi)*e^3*erf(sqrt(e*x + d)*sqrt(-b*c*log(F)/e))/F^((b*c*d - a*c*e)
/e) + 2*(8*(b^3*c^3*e^3*x^3 + 3*b^3*c^3*d*e^2*x^2 + 3*b^3*c^3*d^2*e*x + b^3*c^3*
d^3)*log(F)^3 - 105*e^3 - 28*(b^2*c^2*e^3*x^2 + 2*b^2*c^2*d*e^2*x + b^2*c^2*d^2*
e)*log(F)^2 + 70*(b*c*e^3*x + b*c*d*e^2)*log(F))*sqrt(e*x + d)*sqrt(-b*c*log(F)/
e)*F^(b*c*x + a*c))/(sqrt(-b*c*log(F)/e)*b^4*c^4*log(F)^4)

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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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29598, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)*F^((b*x + a)*c),x, algorithm="giac")

[Out]

Done