3.144 \(\int \frac{x^3 (a+b \log (c x^n))}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=217 \[ -\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac{64 b d^3 n \sqrt{d+e x}}{35 e^4}-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}-\frac{64 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4} \]

[Out]

(64*b*d^3*n*Sqrt[d + e*x])/(35*e^4) - (76*b*d^2*n*(d + e*x)^(3/2))/(105*e^4) + (64*b*d*n*(d + e*x)^(5/2))/(175
*e^4) - (4*b*n*(d + e*x)^(7/2))/(49*e^4) - (64*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(35*e^4) - (2*d^3*S
qrt[d + e*x]*(a + b*Log[c*x^n]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/e^4 - (6*d*(d + e*x)^(5/2)*(
a + b*Log[c*x^n]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4)

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Rubi [A]  time = 0.203931, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {43, 2350, 12, 1620, 50, 63, 208} \[ -\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac{64 b d^3 n \sqrt{d+e x}}{35 e^4}-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}-\frac{64 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*Log[c*x^n]))/Sqrt[d + e*x],x]

[Out]

(64*b*d^3*n*Sqrt[d + e*x])/(35*e^4) - (76*b*d^2*n*(d + e*x)^(3/2))/(105*e^4) + (64*b*d*n*(d + e*x)^(5/2))/(175
*e^4) - (4*b*n*(d + e*x)^(7/2))/(49*e^4) - (64*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(35*e^4) - (2*d^3*S
qrt[d + e*x]*(a + b*Log[c*x^n]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*Log[c*x^n]))/e^4 - (6*d*(d + e*x)^(5/2)*(
a + b*Log[c*x^n]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*Log[c*x^n]))/(7*e^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d+e x}} \, dx &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-(b n) \int \frac{2 \sqrt{d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{35 e^4 x} \, dx\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{(2 b n) \int \frac{\sqrt{d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{x} \, dx}{35 e^4}\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac{(2 b n) \int \left (19 d^2 e \sqrt{d+e x}-\frac{16 d^3 \sqrt{d+e x}}{x}-16 d e (d+e x)^{3/2}+5 e (d+e x)^{5/2}\right ) \, dx}{35 e^4}\\ &=-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac{\left (32 b d^3 n\right ) \int \frac{\sqrt{d+e x}}{x} \, dx}{35 e^4}\\ &=\frac{64 b d^3 n \sqrt{d+e x}}{35 e^4}-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac{\left (32 b d^4 n\right ) \int \frac{1}{x \sqrt{d+e x}} \, dx}{35 e^4}\\ &=\frac{64 b d^3 n \sqrt{d+e x}}{35 e^4}-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}+\frac{\left (64 b d^4 n\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{35 e^5}\\ &=\frac{64 b d^3 n \sqrt{d+e x}}{35 e^4}-\frac{76 b d^2 n (d+e x)^{3/2}}{105 e^4}+\frac{64 b d n (d+e x)^{5/2}}{175 e^4}-\frac{4 b n (d+e x)^{7/2}}{49 e^4}-\frac{64 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{35 e^4}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}\\ \end{align*}

Mathematica [A]  time = 0.219337, size = 150, normalized size = 0.69 \[ -\frac{2 \left (\sqrt{d+e x} \left (105 a \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )+105 b \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right ) \log \left (c x^n\right )+2 b n \left (218 d^2 e x-1276 d^3-111 d e^2 x^2+75 e^3 x^3\right )\right )+3360 b d^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{3675 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/Sqrt[d + e*x],x]

[Out]

(-2*(3360*b*d^(7/2)*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + Sqrt[d + e*x]*(105*a*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2
- 5*e^3*x^3) + 2*b*n*(-1276*d^3 + 218*d^2*e*x - 111*d*e^2*x^2 + 75*e^3*x^3) + 105*b*(16*d^3 - 8*d^2*e*x + 6*d*
e^2*x^2 - 5*e^3*x^3)*Log[c*x^n])))/(3675*e^4)

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Maple [F]  time = 0.605, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))/(e*x+d)^(1/2),x)

[Out]

int(x^3*(a+b*ln(c*x^n))/(e*x+d)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.47126, size = 990, normalized size = 4.56 \begin{align*} \left [\frac{2 \,{\left (1680 \, b d^{\frac{7}{2}} n \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) +{\left (2552 \, b d^{3} n - 1680 \, a d^{3} - 75 \,{\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 6 \,{\left (37 \, b d e^{2} n - 105 \, a d e^{2}\right )} x^{2} - 4 \,{\left (109 \, b d^{2} e n - 210 \, a d^{2} e\right )} x + 105 \,{\left (5 \, b e^{3} x^{3} - 6 \, b d e^{2} x^{2} + 8 \, b d^{2} e x - 16 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (5 \, b e^{3} n x^{3} - 6 \, b d e^{2} n x^{2} + 8 \, b d^{2} e n x - 16 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{3675 \, e^{4}}, \frac{2 \,{\left (3360 \, b \sqrt{-d} d^{3} n \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) +{\left (2552 \, b d^{3} n - 1680 \, a d^{3} - 75 \,{\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 6 \,{\left (37 \, b d e^{2} n - 105 \, a d e^{2}\right )} x^{2} - 4 \,{\left (109 \, b d^{2} e n - 210 \, a d^{2} e\right )} x + 105 \,{\left (5 \, b e^{3} x^{3} - 6 \, b d e^{2} x^{2} + 8 \, b d^{2} e x - 16 \, b d^{3}\right )} \log \left (c\right ) + 105 \,{\left (5 \, b e^{3} n x^{3} - 6 \, b d e^{2} n x^{2} + 8 \, b d^{2} e n x - 16 \, b d^{3} n\right )} \log \left (x\right )\right )} \sqrt{e x + d}\right )}}{3675 \, e^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[2/3675*(1680*b*d^(7/2)*n*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + (2552*b*d^3*n - 1680*a*d^3 - 75*(2*b*
e^3*n - 7*a*e^3)*x^3 + 6*(37*b*d*e^2*n - 105*a*d*e^2)*x^2 - 4*(109*b*d^2*e*n - 210*a*d^2*e)*x + 105*(5*b*e^3*x
^3 - 6*b*d*e^2*x^2 + 8*b*d^2*e*x - 16*b*d^3)*log(c) + 105*(5*b*e^3*n*x^3 - 6*b*d*e^2*n*x^2 + 8*b*d^2*e*n*x - 1
6*b*d^3*n)*log(x))*sqrt(e*x + d))/e^4, 2/3675*(3360*b*sqrt(-d)*d^3*n*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2552*
b*d^3*n - 1680*a*d^3 - 75*(2*b*e^3*n - 7*a*e^3)*x^3 + 6*(37*b*d*e^2*n - 105*a*d*e^2)*x^2 - 4*(109*b*d^2*e*n -
210*a*d^2*e)*x + 105*(5*b*e^3*x^3 - 6*b*d*e^2*x^2 + 8*b*d^2*e*x - 16*b*d^3)*log(c) + 105*(5*b*e^3*n*x^3 - 6*b*
d*e^2*n*x^2 + 8*b*d^2*e*n*x - 16*b*d^3*n)*log(x))*sqrt(e*x + d))/e^4]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{\sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^3/sqrt(e*x + d), x)