Optimal. Leaf size=217 \[ -\frac {2 d^3 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{7 e^4}-\frac {64 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{35 e^4}+\frac {64 b d^3 n \sqrt {d+e x}}{35 e^4}-\frac {76 b d^2 n (d+e x)^{3/2}}{105 e^4}+\frac {64 b d n (d+e x)^{5/2}}{175 e^4}-\frac {4 b n (d+e x)^{7/2}}{49 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 217, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 12
Rule 43
Rule 50
Rule 63
Rule 208
Rule 1620
Rule 2350
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 = 0.23, size = 150, normalized size = 0.69 \[ -\frac {2 \left (\sqrt {d+e x} \left (105 a \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )+105 b \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right ) \log \left (c x^n\right )+2 b n \left (-1276 d^3+218 d^2 e x-111 d e^2 x^2+75 e^3 x^3\right )\right )+3360 b d^{7/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{3675 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 395, normalized size = 1.82 \[ \left [\frac {2 \, {\left (1680 \, b d^{\frac {7}{2}} n \log \left (\frac {e x - 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (2552 \, b d^{3} n - 1680 \, a d^{3} - 75 \, {\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 6 \, {\left (37 \, b d e^{2} n - 105 \, a d e^{2}\right )} x^{2} - 4 \, {\left (109 \, b d^{2} e n - 210 \, a d^{2} e\right )} x + 105 \, {\left (5 \, b e^{3} x^{3} - 6 \, b d e^{2} x^{2} + 8 \, b d^{2} e x - 16 \, b d^{3}\right )} \log \relax (c) + 105 \, {\left (5 \, b e^{3} n x^{3} - 6 \, b d e^{2} n x^{2} + 8 \, b d^{2} e n x - 16 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{3675 \, e^{4}}, \frac {2 \, {\left (3360 \, b \sqrt {-d} d^{3} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (2552 \, b d^{3} n - 1680 \, a d^{3} - 75 \, {\left (2 \, b e^{3} n - 7 \, a e^{3}\right )} x^{3} + 6 \, {\left (37 \, b d e^{2} n - 105 \, a d e^{2}\right )} x^{2} - 4 \, {\left (109 \, b d^{2} e n - 210 \, a d^{2} e\right )} x + 105 \, {\left (5 \, b e^{3} x^{3} - 6 \, b d e^{2} x^{2} + 8 \, b d^{2} e x - 16 \, b d^{3}\right )} \log \relax (c) + 105 \, {\left (5 \, b e^{3} n x^{3} - 6 \, b d e^{2} n x^{2} + 8 \, b d^{2} e n x - 16 \, b d^{3} n\right )} \log \relax (x)\right )} \sqrt {e x + d}\right )}}{3675 \, e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.83, size = 275, normalized size = 1.27 \[ \frac {64 \, b d^{4} n \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right ) e^{\left (-4\right )}}{35 \, \sqrt {-d}} + \frac {2}{3675} \, {\left (525 \, {\left (x e + d\right )}^{\frac {7}{2}} b n \log \left (x e\right ) - 2205 \, {\left (x e + d\right )}^{\frac {5}{2}} b d n \log \left (x e\right ) + 3675 \, {\left (x e + d\right )}^{\frac {3}{2}} b d^{2} n \log \left (x e\right ) - 3675 \, \sqrt {x e + d} b d^{3} n \log \left (x e\right ) - 675 \, {\left (x e + d\right )}^{\frac {7}{2}} b n + 2877 \, {\left (x e + d\right )}^{\frac {5}{2}} b d n - 5005 \, {\left (x e + d\right )}^{\frac {3}{2}} b d^{2} n + 7035 \, \sqrt {x e + d} b d^{3} n + 525 \, {\left (x e + d\right )}^{\frac {7}{2}} b \log \relax (c) - 2205 \, {\left (x e + d\right )}^{\frac {5}{2}} b d \log \relax (c) + 3675 \, {\left (x e + d\right )}^{\frac {3}{2}} b d^{2} \log \relax (c) - 3675 \, \sqrt {x e + d} b d^{3} \log \relax (c) + 525 \, {\left (x e + d\right )}^{\frac {7}{2}} a - 2205 \, {\left (x e + d\right )}^{\frac {5}{2}} a d + 3675 \, {\left (x e + d\right )}^{\frac {3}{2}} a d^{2} - 3675 \, \sqrt {x e + d} a d^{3}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{3}}{\sqrt {e x +d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.45, size = 215, normalized size = 0.99 \[ \frac {4}{3675} \, b n {\left (\frac {840 \, d^{\frac {7}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{4}} - \frac {75 \, {\left (e x + d\right )}^{\frac {7}{2}} - 336 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 665 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 1680 \, \sqrt {e x + d} d^{3}}{e^{4}}\right )} + \frac {2}{35} \, b {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}}}{e^{4}} - \frac {21 \, {\left (e x + d\right )}^{\frac {5}{2}} d}{e^{4}} + \frac {35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2}}{e^{4}} - \frac {35 \, \sqrt {e x + d} d^{3}}{e^{4}}\right )} \log \left (c x^{n}\right ) + \frac {2}{35} \, a {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}}}{e^{4}} - \frac {21 \, {\left (e x + d\right )}^{\frac {5}{2}} d}{e^{4}} + \frac {35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2}}{e^{4}} - \frac {35 \, \sqrt {e x + d} d^{3}}{e^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________