3.35 \(\int \frac{f^{a+3 b x}}{c+d f^{e+2 b x}} \, dx\)

Optimal. Leaf size=88 \[ \frac{f^{\frac{1}{2} (2 a-3 e)+\frac{1}{2} (2 b x+e)}}{b d \log (f)}-\frac{\sqrt{c} f^{a-\frac{3 e}{2}} \tan ^{-1}\left (\frac{\sqrt{d} f^{\frac{1}{2} (2 b x+e)}}{\sqrt{c}}\right )}{b d^{3/2} \log (f)} \]

[Out]

f^((2*a - 3*e)/2 + (e + 2*b*x)/2)/(b*d*Log[f]) - (Sqrt[c]*f^(a - (3*e)/2)*ArcTan
[(Sqrt[d]*f^((e + 2*b*x)/2))/Sqrt[c]])/(b*d^(3/2)*Log[f])

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Rubi [A]  time = 0.128457, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{f^{\frac{1}{2} (2 a-3 e)+\frac{1}{2} (2 b x+e)}}{b d \log (f)}-\frac{\sqrt{c} f^{a-\frac{3 e}{2}} \tan ^{-1}\left (\frac{\sqrt{d} f^{\frac{1}{2} (2 b x+e)}}{\sqrt{c}}\right )}{b d^{3/2} \log (f)} \]

Antiderivative was successfully verified.

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

[Out]

f^((2*a - 3*e)/2 + (e + 2*b*x)/2)/(b*d*Log[f]) - (Sqrt[c]*f^(a - (3*e)/2)*ArcTan
[(Sqrt[d]*f^((e + 2*b*x)/2))/Sqrt[c]])/(b*d^(3/2)*Log[f])

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Rubi in Sympy [A]  time = 23.4149, size = 66, normalized size = 0.75 \[ - \frac{\sqrt{c} f^{a - \frac{3 e}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} f^{b x + \frac{e}{2}}}{\sqrt{c}} \right )}}{b d^{\frac{3}{2}} \log{\left (f \right )}} + \frac{f^{a - \frac{3 e}{2}} f^{b x + \frac{e}{2}}}{b d \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(3*b*x+a)/(c+d*f**(2*b*x+e)),x)

[Out]

-sqrt(c)*f**(a - 3*e/2)*atan(sqrt(d)*f**(b*x + e/2)/sqrt(c))/(b*d**(3/2)*log(f))
+ f**(a - 3*e/2)*f**(b*x + e/2)/(b*d*log(f))

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Mathematica [A]  time = 0.0737817, size = 67, normalized size = 0.76 \[ \frac{\frac{f^{a+b x-e}}{d}-\frac{\sqrt{c} f^{a-\frac{3 e}{2}} \tan ^{-1}\left (\frac{\sqrt{d} f^{b x+\frac{e}{2}}}{\sqrt{c}}\right )}{d^{3/2}}}{b \log (f)} \]

Antiderivative was successfully verified.

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

[Out]

(f^(a - e + b*x)/d - (Sqrt[c]*f^(a - (3*e)/2)*ArcTan[(Sqrt[d]*f^(e/2 + b*x))/Sqr
t[c]])/d^(3/2))/(b*Log[f])

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Maple [B]  time = 0.108, size = 171, normalized size = 1.9 \[{\frac{1}{d\ln \left ( f \right ) b}{f}^{bx+{\frac{a}{3}}} \left ({f}^{{\frac{e}{2}}} \right ) ^{-2} \left ({f}^{-{\frac{a}{3}}} \right ) ^{-2}}+{\frac{1}{2\,b{d}^{2}\ln \left ( f \right ) }\sqrt{-cd}\ln \left ({f}^{bx+{\frac{a}{3}}}-{\frac{1}{d}\sqrt{-cd} \left ({f}^{{\frac{e}{2}}} \right ) ^{-1} \left ({f}^{-{\frac{a}{3}}} \right ) ^{-1}} \right ) \left ({f}^{-{\frac{a}{3}}} \right ) ^{-3} \left ({f}^{{\frac{e}{2}}} \right ) ^{-3}}-{\frac{1}{2\,b{d}^{2}\ln \left ( f \right ) }\sqrt{-cd}\ln \left ({f}^{bx+{\frac{a}{3}}}+{\frac{1}{d}\sqrt{-cd} \left ({f}^{{\frac{e}{2}}} \right ) ^{-1} \left ({f}^{-{\frac{a}{3}}} \right ) ^{-1}} \right ) \left ({f}^{-{\frac{a}{3}}} \right ) ^{-3} \left ({f}^{{\frac{e}{2}}} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(3*b*x+a)/(c+d*f^(2*b*x+e)),x)

[Out]

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

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Maxima [A]  time = 0.958958, size = 171, normalized size = 1.94 \[ -\frac{c f^{a - e} \log \left (\frac{d{\left (f^{3 \, b x + a}\right )}^{\frac{1}{3}} f^{e} - \sqrt{-c d f^{e}} f^{\frac{1}{3} \, a}}{d{\left (f^{3 \, b x + a}\right )}^{\frac{1}{3}} f^{e} + \sqrt{-c d f^{e}} f^{\frac{1}{3} \, a}}\right )}{2 \, \sqrt{-c d f^{e}} b d \log \left (f\right )} + \frac{{\left (f^{3 \, b x + a}\right )}^{\frac{1}{3}} f^{\frac{2}{3} \, a - e}}{b d \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(3*b*x + a)/(d*f^(2*b*x + e) + c),x, algorithm="maxima")

[Out]

-1/2*c*f^(a - e)*log((d*(f^(3*b*x + a))^(1/3)*f^e - sqrt(-c*d*f^e)*f^(1/3*a))/(d
*(f^(3*b*x + a))^(1/3)*f^e + sqrt(-c*d*f^e)*f^(1/3*a)))/(sqrt(-c*d*f^e)*b*d*log(
f)) + (f^(3*b*x + a))^(1/3)*f^(2/3*a - e)/(b*d*log(f))

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Fricas [A]  time = 0.264311, size = 1, normalized size = 0.01 \[ \left [\frac{f^{a - \frac{3}{2} \, e} \sqrt{-\frac{c}{d}} \log \left (-\frac{2 \, d f^{b x + \frac{1}{2} \, e} \sqrt{-\frac{c}{d}} - d f^{2 \, b x + e} + c}{d f^{2 \, b x + e} + c}\right ) + 2 \, f^{b x + \frac{1}{2} \, e} f^{a - \frac{3}{2} \, e}}{2 \, b d \log \left (f\right )}, -\frac{f^{a - \frac{3}{2} \, e} \sqrt{\frac{c}{d}} \arctan \left (\frac{f^{b x + \frac{1}{2} \, e}}{\sqrt{\frac{c}{d}}}\right ) - f^{b x + \frac{1}{2} \, e} f^{a - \frac{3}{2} \, e}}{b d \log \left (f\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(f^(a - 3/2*e)*sqrt(-c/d)*log(-(2*d*f^(b*x + 1/2*e)*sqrt(-c/d) - d*f^(2*b*x
+ e) + c)/(d*f^(2*b*x + e) + c)) + 2*f^(b*x + 1/2*e)*f^(a - 3/2*e))/(b*d*log(f)
), -(f^(a - 3/2*e)*sqrt(c/d)*arctan(f^(b*x + 1/2*e)/sqrt(c/d)) - f^(b*x + 1/2*e)
*f^(a - 3/2*e))/(b*d*log(f))]

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Sympy [A]  time = 3.67459, size = 253, normalized size = 2.88 \[ \begin{cases} \frac{e^{\frac{2 a \log{\left (f \right )}}{3}} e^{- e \log{\left (f \right )}} e^{\frac{\left (a + 3 b x\right ) \log{\left (f \right )}}{3}}}{b d \log{\left (f \right )}} & \text{for}\: b d e^{e \log{\left (f \right )}} \log{\left (f \right )} \neq 0 \\\frac{x \left (c^{2} e^{\frac{10 a \log{\left (f \right )}}{3}} + 2 c d e^{\frac{8 a \log{\left (f \right )}}{3}} e^{e \log{\left (f \right )}} + d^{2} e^{2 a \log{\left (f \right )}} e^{2 e \log{\left (f \right )}}\right )}{c^{2} d e^{\frac{8 a \log{\left (f \right )}}{3}} e^{e \log{\left (f \right )}} + 2 c d^{2} e^{2 a \log{\left (f \right )}} e^{2 e \log{\left (f \right )}} + d^{3} e^{\frac{4 a \log{\left (f \right )}}{3}} e^{3 e \log{\left (f \right )}}} & \text{otherwise} \end{cases} + \operatorname{RootSum}{\left (4 z^{2} b^{2} d^{3} e^{3 e \log{\left (f \right )}} \log{\left (f \right )}^{2} + c e^{2 a \log{\left (f \right )}}, \left ( i \mapsto i \log{\left (- 2 i b d e^{- \frac{2 a \log{\left (f \right )}}{3}} e^{e \log{\left (f \right )}} \log{\left (f \right )} + e^{\frac{\left (a + 3 b x\right ) \log{\left (f \right )}}{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(3*b*x+a)/(c+d*f**(2*b*x+e)),x)

[Out]

Piecewise((exp(2*a*log(f)/3)*exp(-e*log(f))*exp((a + 3*b*x)*log(f)/3)/(b*d*log(f
)), Ne(b*d*exp(e*log(f))*log(f), 0)), (x*(c**2*exp(10*a*log(f)/3) + 2*c*d*exp(8*
a*log(f)/3)*exp(e*log(f)) + d**2*exp(2*a*log(f))*exp(2*e*log(f)))/(c**2*d*exp(8*
a*log(f)/3)*exp(e*log(f)) + 2*c*d**2*exp(2*a*log(f))*exp(2*e*log(f)) + d**3*exp(
4*a*log(f)/3)*exp(3*e*log(f))), True)) + RootSum(4*_z**2*b**2*d**3*exp(3*e*log(f
))*log(f)**2 + c*exp(2*a*log(f)), Lambda(_i, _i*log(-2*_i*b*d*exp(-2*a*log(f)/3)
*exp(e*log(f))*log(f) + exp((a + 3*b*x)*log(f)/3))))

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GIAC/XCAS [A]  time = 0.240034, size = 104, normalized size = 1.18 \[ -f^{a}{\left (\frac{c \arctan \left (\frac{d f^{b x} f^{e}}{\sqrt{c d f^{e}}}\right )}{\sqrt{c d f^{e}} b d f^{e}{\rm ln}\left (f\right )} - \frac{f^{b x}}{b d f^{e}{\rm ln}\left (f\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(3*b*x + a)/(d*f^(2*b*x + e) + c),x, algorithm="giac")

[Out]

-f^a*(c*arctan(d*f^(b*x)*f^e/sqrt(c*d*f^e))/(sqrt(c*d*f^e)*b*d*f^e*ln(f)) - f^(b
*x)/(b*d*f^e*ln(f)))