3.135 \(\int \frac {f+g x}{(a+b \log (c (d+e x)^n))^{5/2}} \, dx\)

Optimal. Leaf size=311 \[ \frac {4 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}+\frac {8 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}+\frac {4 (d+e x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

[Out]

-2/3*(e*x+d)*(g*x+f)/b/e/n/(a+b*ln(c*(e*x+d)^n))^(3/2)+4/3*(-d*g+e*f)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)
/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(5/2)/e^2/exp(a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(1/n))+8/3*g*(e*x+d)^2*erfi(2^(1/2)*(
a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)/e^2/exp(2*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(
2/n))+4/3*(-d*g+e*f)*(e*x+d)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))^(1/2)-8/3*(e*x+d)*(g*x+f)/b^2/e/n^2/(a+b*ln(c*(
e*x+d)^n))^(1/2)

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Rubi [A]  time = 0.56, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2297} \[ \frac {4 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}+\frac {8 \sqrt {2 \pi } g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}+\frac {4 (d+e x) (e f-d g)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/(a + b*Log[c*(d + e*x)^n])^(5/2),x]

[Out]

(4*(e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^2*E^(a/
(b*n))*n^(5/2)*(c*(d + e*x)^n)^n^(-1)) + (8*g*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^
n]])/(Sqrt[b]*Sqrt[n])])/(3*b^(5/2)*e^2*E^((2*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(2/n)) - (2*(d + e*x)*(f + g*x
))/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) + (4*(e*f - d*g)*(d + e*x))/(3*b^2*e^2*n^2*Sqrt[a + b*Log[c*(d +
e*x)^n]]) - (8*(d + e*x)*(f + g*x))/(3*b^2*e*n^2*Sqrt[a + b*Log[c*(d + e*x)^n]])

Rule 2180

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] &&  !$UseGamma === True

Rule 2204

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]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2400

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
d + e*x)*(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1))/(b*e*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[(q*(e*f - d*g))/(b*e*n*(p + 1)), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx &=-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b n}-\frac {(2 (e f-d g)) \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx}{3 b e n}\\ &=-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {16 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 n^2}-\frac {(8 (e f-d g)) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}-\frac {(2 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^{3/2}} \, dx,x,d+e x\right )}{3 b e^2 n}\\ &=-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {16 \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx}{3 b^2 n^2}-\frac {(4 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^2 n^2}-\frac {(8 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^2 n^2}\\ &=-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {(16 g) \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}+\frac {(16 (e f-d g)) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 b^2 e n^2}-\frac {\left (4 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^2 n^3}-\frac {\left (8 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^2 n^3}\\ &=-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {(16 g) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^2 n^2}+\frac {(16 (e f-d g)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{3 b^2 e^2 n^2}-\frac {\left (8 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^2 n^3}-\frac {\left (16 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^2 n^3}\\ &=-\frac {4 e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^2 n^{5/2}}-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (16 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^2 n^3}+\frac {\left (16 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{3 b^2 e^2 n^3}\\ &=-\frac {4 e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^2 n^{5/2}}-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (32 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b n}+\frac {2 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^2 n^3}+\frac {\left (32 (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{3 b^3 e^2 n^3}\\ &=\frac {4 e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}+\frac {8 e^{-\frac {2 a}{b n}} g \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^2 n^{5/2}}-\frac {2 (d+e x) (f+g x)}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {4 (e f-d g) (d+e x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {8 (d+e x) (f+g x)}{3 b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\\ \end {align*}

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Mathematica [A]  time = 1.96, size = 353, normalized size = 1.14 \[ \frac {2 e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-8 \sqrt {\pi } d g e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+4 \sqrt {2 \pi } g (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {\sqrt {b} \sqrt {n} e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (2 a (d g+e f+2 e g x)+2 b (d g+e (f+2 g x)) \log \left (c (d+e x)^n\right )+b e n (f+g x)\right )+2 b n (3 d g+e f) \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\right )}{3 b^{5/2} e^2 n^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/(a + b*Log[c*(d + e*x)^n])^(5/2),x]

[Out]

(2*(d + e*x)*(-8*d*E^(a/(b*n))*g*Sqrt[Pi]*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*
Sqrt[n])] + 4*g*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])] - (Sqrt[
b]*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)*(2*b*(e*f + 3*d*g)*n*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*
n))]*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^(3/2) + E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(b*e*n*(f + g*x) + 2*a*(
e*f + d*g + 2*e*g*x) + 2*b*(d*g + e*(f + 2*g*x))*Log[c*(d + e*x)^n])))/(a + b*Log[c*(d + e*x)^n])^(3/2)))/(3*b
^(5/2)*e^2*E^((2*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(2/n))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)/(b*log((e*x + d)^n*c) + a)^(5/2), x)

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maple [F]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {g x +f}{\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(b*ln(c*(e*x+d)^n)+a)^(5/2),x)

[Out]

int((g*x+f)/(b*ln(c*(e*x+d)^n)+a)^(5/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n))^(5/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)/(b*log((e*x + d)^n*c) + a)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {f+g\,x}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)/(a + b*log(c*(d + e*x)^n))^(5/2),x)

[Out]

int((f + g*x)/(a + b*log(c*(d + e*x)^n))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f + g x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*ln(c*(e*x+d)**n))**(5/2),x)

[Out]

Integral((f + g*x)/(a + b*log(c*(d + e*x)**n))**(5/2), x)

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