Integrand size = 23, antiderivative size = 146 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\frac {20 b d n \sqrt {d+e x}}{3 e^3}-\frac {4 b n (d+e x)^{3/2}}{9 e^3}-\frac {32 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{3 e^3}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3} \]
-4/9*b*n*(e*x+d)^(3/2)/e^3-32/3*b*d^(3/2)*n*arctanh((e*x+d)^(1/2)/d^(1/2)) /e^3+2/3*(e*x+d)^(3/2)*(a+b*ln(c*x^n))/e^3-2*d^2*(a+b*ln(c*x^n))/e^3/(e*x+ d)^(1/2)+20/3*b*d*n*(e*x+d)^(1/2)/e^3-4*d*(a+b*ln(c*x^n))*(e*x+d)^(1/2)/e^ 3
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\frac {-48 a d^2+56 b d^2 n-24 a d e x+52 b d e n x+6 a e^2 x^2-4 b e^2 n x^2-96 b d^{3/2} n \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-6 b \left (8 d^2+4 d e x-e^2 x^2\right ) \log \left (c x^n\right )}{9 e^3 \sqrt {d+e x}} \]
(-48*a*d^2 + 56*b*d^2*n - 24*a*d*e*x + 52*b*d*e*n*x + 6*a*e^2*x^2 - 4*b*e^ 2*n*x^2 - 96*b*d^(3/2)*n*Sqrt[d + e*x]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] - 6* b*(8*d^2 + 4*d*e*x - e^2*x^2)*Log[c*x^n])/(9*e^3*Sqrt[d + e*x])
Time = 0.40 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2792, 27, 1192, 25, 1467, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2792 |
| \(\displaystyle -b n \int -\frac {2 \left (8 d^2+4 e x d-e^2 x^2\right )}{3 e^3 x \sqrt {d+e x}}dx-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b n \int \frac {8 d^2+4 e x d-e^2 x^2}{x \sqrt {d+e x}}dx}{3 e^3}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {4 b n \int \frac {3 d^2 e^2-(d+e x)^2 e^2+6 d (d+e x) e^2}{e x}d\sqrt {d+e x}}{3 e^5}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 b n \int -\frac {3 d^2 e^2-(d+e x)^2 e^2+6 d (d+e x) e^2}{e x}d\sqrt {d+e x}}{3 e^5}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {4 b n \int \left (-\frac {8 e d^2}{x}-5 e^2 d+e^2 (d+e x)\right )d\sqrt {d+e x}}{3 e^5}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b n \left (-8 d^{3/2} e^2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\frac {1}{3} e^2 (d+e x)^{3/2}+5 d e^2 \sqrt {d+e x}\right )}{3 e^5}\) |
(4*b*n*(5*d*e^2*Sqrt[d + e*x] - (e^2*(d + e*x)^(3/2))/3 - 8*d^(3/2)*e^2*Ar cTanh[Sqrt[d + e*x]/Sqrt[d]]))/(3*e^5) - (2*d^2*(a + b*Log[c*x^n]))/(e^3*S qrt[d + e*x]) - (4*d*Sqrt[d + e*x]*(a + b*Log[c*x^n]))/e^3 + (2*(d + e*x)^ (3/2)*(a + b*Log[c*x^n]))/(3*e^3)
3.2.52.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[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])
\[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e x +d \right )^{\frac {3}{2}}}d x\]
Time = 0.32 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.26 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\left [\frac {2 \, {\left (24 \, {\left (b d e n x + b d^{2} n\right )} \sqrt {d} \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (28 \, b d^{2} n - 24 \, a d^{2} - {\left (2 \, b e^{2} n - 3 \, a e^{2}\right )} x^{2} + 2 \, {\left (13 \, b d e n - 6 \, a d e\right )} x + 3 \, {\left (b e^{2} x^{2} - 4 \, b d e x - 8 \, b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (b e^{2} n x^{2} - 4 \, b d e n x - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{9 \, {\left (e^{4} x + d e^{3}\right )}}, \frac {2 \, {\left (48 \, {\left (b d e n x + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (28 \, b d^{2} n - 24 \, a d^{2} - {\left (2 \, b e^{2} n - 3 \, a e^{2}\right )} x^{2} + 2 \, {\left (13 \, b d e n - 6 \, a d e\right )} x + 3 \, {\left (b e^{2} x^{2} - 4 \, b d e x - 8 \, b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (b e^{2} n x^{2} - 4 \, b d e n x - 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{9 \, {\left (e^{4} x + d e^{3}\right )}}\right ] \]
[2/9*(24*(b*d*e*n*x + b*d^2*n)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + (28*b*d^2*n - 24*a*d^2 - (2*b*e^2*n - 3*a*e^2)*x^2 + 2*(13*b*d *e*n - 6*a*d*e)*x + 3*(b*e^2*x^2 - 4*b*d*e*x - 8*b*d^2)*log(c) + 3*(b*e^2* n*x^2 - 4*b*d*e*n*x - 8*b*d^2*n)*log(x))*sqrt(e*x + d))/(e^4*x + d*e^3), 2 /9*(48*(b*d*e*n*x + b*d^2*n)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ( 28*b*d^2*n - 24*a*d^2 - (2*b*e^2*n - 3*a*e^2)*x^2 + 2*(13*b*d*e*n - 6*a*d* e)*x + 3*(b*e^2*x^2 - 4*b*d*e*x - 8*b*d^2)*log(c) + 3*(b*e^2*n*x^2 - 4*b*d *e*n*x - 8*b*d^2*n)*log(x))*sqrt(e*x + d))/(e^4*x + d*e^3)]
Time = 135.75 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.11 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=a \left (\begin {cases} - \frac {2 d^{2}}{e^{3} \sqrt {d + e x}} - \frac {4 d \sqrt {d + e x}}{e^{3}} + \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} & \text {for}\: e \neq 0 \\\frac {x^{3}}{3 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {16 d^{\frac {3}{2}} \sqrt {1 + \frac {e x}{d}}}{9 e^{3}} + \frac {2 d^{\frac {3}{2}} \log {\left (\frac {e x}{d} \right )}}{3 e^{3}} - \frac {4 d^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{3 e^{3}} + \frac {12 d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} \sqrt {x}} \right )}}{e^{3}} + \frac {4 \sqrt {d} x \sqrt {1 + \frac {e x}{d}}}{9 e^{2}} - \frac {8 d^{2}}{e^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {d}{e x} + 1}} - \frac {8 d \sqrt {x}}{e^{\frac {5}{2}} \sqrt {\frac {d}{e x} + 1}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x^{3}}{9 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} - \frac {2 d^{2}}{e^{3} \sqrt {d + e x}} - \frac {4 d \sqrt {d + e x}}{e^{3}} + \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e^{3}} & \text {for}\: e \neq 0 \\\frac {x^{3}}{3 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
a*Piecewise((-2*d**2/(e**3*sqrt(d + e*x)) - 4*d*sqrt(d + e*x)/e**3 + 2*(d + e*x)**(3/2)/(3*e**3), Ne(e, 0)), (x**3/(3*d**(3/2)), True)) - b*n*Piecew ise((16*d**(3/2)*sqrt(1 + e*x/d)/(9*e**3) + 2*d**(3/2)*log(e*x/d)/(3*e**3) - 4*d**(3/2)*log(sqrt(1 + e*x/d) + 1)/(3*e**3) + 12*d**(3/2)*asinh(sqrt(d )/(sqrt(e)*sqrt(x)))/e**3 + 4*sqrt(d)*x*sqrt(1 + e*x/d)/(9*e**2) - 8*d**2/ (e**(7/2)*sqrt(x)*sqrt(d/(e*x) + 1)) - 8*d*sqrt(x)/(e**(5/2)*sqrt(d/(e*x) + 1)), (e > -oo) & (e < oo) & Ne(e, 0)), (x**3/(9*d**(3/2)), True)) + b*Pi ecewise((-2*d**2/(e**3*sqrt(d + e*x)) - 4*d*sqrt(d + e*x)/e**3 + 2*(d + e* x)**(3/2)/(3*e**3), Ne(e, 0)), (x**3/(3*d**(3/2)), True))*log(c*x**n)
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\frac {4}{9} \, b n {\left (\frac {12 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{3}} - \frac {{\left (e x + d\right )}^{\frac {3}{2}} - 15 \, \sqrt {e x + d} d}{e^{3}}\right )} + \frac {2}{3} \, b {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{e^{3}} - \frac {6 \, \sqrt {e x + d} d}{e^{3}} - \frac {3 \, d^{2}}{\sqrt {e x + d} e^{3}}\right )} \log \left (c x^{n}\right ) + \frac {2}{3} \, a {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{e^{3}} - \frac {6 \, \sqrt {e x + d} d}{e^{3}} - \frac {3 \, d^{2}}{\sqrt {e x + d} e^{3}}\right )} \]
4/9*b*n*(12*d^(3/2)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d) ))/e^3 - ((e*x + d)^(3/2) - 15*sqrt(e*x + d)*d)/e^3) + 2/3*b*((e*x + d)^(3 /2)/e^3 - 6*sqrt(e*x + d)*d/e^3 - 3*d^2/(sqrt(e*x + d)*e^3))*log(c*x^n) + 2/3*a*((e*x + d)^(3/2)/e^3 - 6*sqrt(e*x + d)*d/e^3 - 3*d^2/(sqrt(e*x + d)* e^3))
\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]