3.5.2 \(\int x (a+b \log (c (d+e \sqrt {x})^n)) \, dx\) [402]

Optimal. Leaf size=102 \[ \frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]

[Out]

-1/4*b*d^2*n*x/e^2+1/6*b*d*n*x^(3/2)/e-1/8*b*n*x^2-1/2*b*d^4*n*ln(d+e*x^(1/2))/e^4+1/2*x^2*(a+b*ln(c*(d+e*x^(1
/2))^n))+1/2*b*d^3*n*x^(1/2)/e^3

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Rubi [A]
time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2504, 2442, 45} \begin {gather*} \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^3*n*Sqrt[x])/(2*e^3) - (b*d^2*n*x)/(4*e^2) + (b*d*n*x^(3/2))/(6*e) - (b*n*x^2)/8 - (b*d^4*n*Log[d + e*Sqr
t[x]])/(2*e^4) + (x^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/2

Rule 45

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 2442

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

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx &=2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^4}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 107, normalized size = 1.05 \begin {gather*} \frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}+\frac {a x^2}{2}-\frac {1}{8} b n x^2-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {1}{2} b x^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^3*n*Sqrt[x])/(2*e^3) - (b*d^2*n*x)/(4*e^2) + (b*d*n*x^(3/2))/(6*e) + (a*x^2)/2 - (b*n*x^2)/8 - (b*d^4*n*L
og[d + e*Sqrt[x]])/(2*e^4) + (b*x^2*Log[c*(d + e*Sqrt[x])^n])/2

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 84, normalized size = 0.82 \begin {gather*} -\frac {1}{24} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + {\left (6 \, d^{2} x e - 12 \, d^{3} \sqrt {x} - 4 \, d x^{\frac {3}{2}} e^{2} + 3 \, x^{2} e^{3}\right )} e^{\left (-4\right )}\right )} b n e + \frac {1}{2} \, b x^{2} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*d^4*e^(-5)*log(sqrt(x)*e + d) + (6*d^2*x*e - 12*d^3*sqrt(x) - 4*d*x^(3/2)*e^2 + 3*x^2*e^3)*e^(-4))*b
*n*e + 1/2*b*x^2*log((sqrt(x)*e + d)^n*c) + 1/2*a*x^2

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Fricas [A]
time = 0.38, size = 88, normalized size = 0.86 \begin {gather*} -\frac {1}{24} \, {\left (6 \, b d^{2} n x e^{2} - 12 \, b x^{2} e^{4} \log \left (c\right ) + 3 \, {\left (b n - 4 \, a\right )} x^{2} e^{4} + 12 \, {\left (b d^{4} n - b n x^{2} e^{4}\right )} \log \left (\sqrt {x} e + d\right ) - 4 \, {\left (3 \, b d^{3} n e + b d n x e^{3}\right )} \sqrt {x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(6*b*d^2*n*x*e^2 - 12*b*x^2*e^4*log(c) + 3*(b*n - 4*a)*x^2*e^4 + 12*(b*d^4*n - b*n*x^2*e^4)*log(sqrt(x)*
e + d) - 4*(3*b*d^3*n*e + b*d*n*x*e^3)*sqrt(x))*e^(-4)

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Sympy [A]
time = 1.90, size = 100, normalized size = 0.98 \begin {gather*} \frac {a x^{2}}{2} + b \left (- \frac {e n \left (\frac {2 d^{4} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{4}} - \frac {2 d^{3} \sqrt {x}}{e^{4}} + \frac {d^{2} x}{e^{3}} - \frac {2 d x^{\frac {3}{2}}}{3 e^{2}} + \frac {x^{2}}{2 e}\right )}{4} + \frac {x^{2} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*(-e*n*(2*d**4*Piecewise((sqrt(x)/d, Eq(e, 0)), (log(d + e*sqrt(x))/e, True))/e**4 - 2*d**3*sqrt(x
)/e**4 + d**2*x/e**3 - 2*d*x**(3/2)/(3*e**2) + x**2/(2*e))/4 + x**2*log(c*(d + e*sqrt(x))**n)/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (80) = 160\).
time = 4.16, size = 185, normalized size = 1.81 \begin {gather*} \frac {1}{24} \, {\left (12 \, b x^{2} e \log \left (c\right ) + 12 \, a x^{2} e + {\left (12 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 72 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 3 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} + 16 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} - 36 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} + 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )}\right )} b n\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*b*x^2*e*log(c) + 12*a*x^2*e + (12*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*
d*e^(-3)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-
3)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16*(sqrt(x)*e + d)^3*d*e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*
e^(-3) + 48*(sqrt(x)*e + d)*d^3*e^(-3))*b*n)*e^(-1)

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Mupad [B]
time = 0.41, size = 85, normalized size = 0.83 \begin {gather*} \frac {a\,x^2}{2}-\frac {b\,n\,x^2}{8}+\frac {b\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{2}-\frac {b\,d^2\,n\,x}{4\,e^2}+\frac {b\,d\,n\,x^{3/2}}{6\,e}-\frac {b\,d^4\,n\,\ln \left (d+e\,\sqrt {x}\right )}{2\,e^4}+\frac {b\,d^3\,n\,\sqrt {x}}{2\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x^2)/2 - (b*n*x^2)/8 + (b*x^2*log(c*(d + e*x^(1/2))^n))/2 - (b*d^2*n*x)/(4*e^2) + (b*d*n*x^(3/2))/(6*e) - (
b*d^4*n*log(d + e*x^(1/2)))/(2*e^4) + (b*d^3*n*x^(1/2))/(2*e^3)

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