Optimal. Leaf size=169 \[ -\frac {32 b d^2 n \sqrt {d+e x}}{15 e^3}+\frac {28 b d n (d+e x)^{3/2}}{45 e^3}-\frac {4 b n (d+e x)^{5/2}}{25 e^3}+\frac {32 b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{15 e^3}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {45, 2392, 12,
911, 1275, 214} \begin {gather*} \frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}+\frac {32 b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{15 e^3}-\frac {32 b d^2 n \sqrt {d+e x}}{15 e^3}+\frac {28 b d n (d+e x)^{3/2}}{45 e^3}-\frac {4 b n (d+e x)^{5/2}}{25 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 214
Rule 911
Rule 1275
Rule 2392
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}} \, dx &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-(b n) \int \frac {2 \sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{15 e^3 x} \, dx\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {(2 b n) \int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{x} \, dx}{15 e^3}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {(4 b n) \text {Subst}\left (\int \frac {x^2 \left (15 d^2-10 d x^2+3 x^4\right )}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{15 e^4}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {(4 b n) \text {Subst}\left (\int \left (8 d^2 e-7 d e x^2+3 e x^4+\frac {8 d^3}{-\frac {d}{e}+\frac {x^2}{e}}\right ) \, dx,x,\sqrt {d+e x}\right )}{15 e^4}\\ &=-\frac {32 b d^2 n \sqrt {d+e x}}{15 e^3}+\frac {28 b d n (d+e x)^{3/2}}{45 e^3}-\frac {4 b n (d+e x)^{5/2}}{25 e^3}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}-\frac {\left (32 b d^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{15 e^4}\\ &=-\frac {32 b d^2 n \sqrt {d+e x}}{15 e^3}+\frac {28 b d n (d+e x)^{3/2}}{45 e^3}-\frac {4 b n (d+e x)^{5/2}}{25 e^3}+\frac {32 b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{15 e^3}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 118, normalized size = 0.70 \begin {gather*} \frac {480 b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 \sqrt {d+e x} \left (15 a \left (8 d^2-4 d e x+3 e^2 x^2\right )-2 b n \left (94 d^2-17 d e x+9 e^2 x^2\right )+15 b \left (8 d^2-4 d e x+3 e^2 x^2\right ) \log \left (c x^n\right )\right )}{225 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {e x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 175, normalized size = 1.04 \begin {gather*} -\frac {4}{225} \, {\left (60 \, d^{\frac {5}{2}} e^{\left (-3\right )} \log \left (\frac {\sqrt {x e + d} - \sqrt {d}}{\sqrt {x e + d} + \sqrt {d}}\right ) + {\left (9 \, {\left (x e + d\right )}^{\frac {5}{2}} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 120 \, \sqrt {x e + d} d^{2}\right )} e^{\left (-3\right )}\right )} b n + \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} e^{\left (-3\right )} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d e^{\left (-3\right )} + 15 \, \sqrt {x e + d} d^{2} e^{\left (-3\right )}\right )} b \log \left (c x^{n}\right ) + \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} e^{\left (-3\right )} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d e^{\left (-3\right )} + 15 \, \sqrt {x e + d} d^{2} e^{\left (-3\right )}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 291, normalized size = 1.72 \begin {gather*} \left [\frac {2}{225} \, {\left (120 \, b d^{\frac {5}{2}} n \log \left (\frac {x e + 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (188 \, b d^{2} n + 9 \, {\left (2 \, b n - 5 \, a\right )} x^{2} e^{2} - 120 \, a d^{2} - 2 \, {\left (17 \, b d n - 30 \, a d\right )} x e - 15 \, {\left (3 \, b x^{2} e^{2} - 4 \, b d x e + 8 \, b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (3 \, b n x^{2} e^{2} - 4 \, b d n x e + 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )}, -\frac {2}{225} \, {\left (240 \, b \sqrt {-d} d^{2} n \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (188 \, b d^{2} n + 9 \, {\left (2 \, b n - 5 \, a\right )} x^{2} e^{2} - 120 \, a d^{2} - 2 \, {\left (17 \, b d n - 30 \, a d\right )} x e - 15 \, {\left (3 \, b x^{2} e^{2} - 4 \, b d x e + 8 \, b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (3 \, b n x^{2} e^{2} - 4 \, b d n x e + 8 \, b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 714 vs.
\(2 (170) = 340\).
time = 122.51, size = 714, normalized size = 4.22 \begin {gather*} \begin {cases} \frac {- \frac {2 a d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e^{2}} - \frac {2 a \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} - \frac {2 b d \left (d^{2} \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) - 2 d \left (- \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right ) - \frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (- d e \sqrt {d + e x} - \frac {d e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right )}{e^{2}} - \frac {2 b \left (- d^{3} \left (\frac {\log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{\sqrt {d + e x}} - \frac {2 n \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{d \sqrt {- \frac {1}{d}}}\right ) + 3 d^{2} \left (- \sqrt {d + e x} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )} - \frac {2 n \left (- e \sqrt {d + e x} - \frac {e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}}\right )}{e}\right ) - 3 d \left (- \frac {\left (d + e x\right )^{\frac {3}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{3} - \frac {2 n \left (- d e \sqrt {d + e x} - \frac {d e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {e \left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{3 e}\right ) - \frac {\left (d + e x\right )^{\frac {5}{2}} \log {\left (c \left (- \frac {d}{e} + \frac {d + e x}{e}\right )^{n} \right )}}{5} - \frac {2 n \left (- d^{2} e \sqrt {d + e x} - \frac {d^{2} e \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x}} \right )}}{\sqrt {- \frac {1}{d}}} - \frac {d e \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {e \left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{5 e}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {\frac {a x^{3}}{3} + b \left (- \frac {n x^{3}}{9} + \frac {x^{3} \log {\left (c x^{n} \right )}}{3}\right )}{\sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.97, size = 210, normalized size = 1.24 \begin {gather*} -\frac {32 \, b d^{3} n \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right ) e^{\left (-3\right )}}{15 \, \sqrt {-d}} + \frac {2}{225} \, {\left (45 \, {\left (x e + d\right )}^{\frac {5}{2}} b n \log \left (x e\right ) - 150 \, {\left (x e + d\right )}^{\frac {3}{2}} b d n \log \left (x e\right ) + 225 \, \sqrt {x e + d} b d^{2} n \log \left (x e\right ) - 63 \, {\left (x e + d\right )}^{\frac {5}{2}} b n + 220 \, {\left (x e + d\right )}^{\frac {3}{2}} b d n - 465 \, \sqrt {x e + d} b d^{2} n + 45 \, {\left (x e + d\right )}^{\frac {5}{2}} b \log \left (c\right ) - 150 \, {\left (x e + d\right )}^{\frac {3}{2}} b d \log \left (c\right ) + 225 \, \sqrt {x e + d} b d^{2} \log \left (c\right ) + 45 \, {\left (x e + d\right )}^{\frac {5}{2}} a - 150 \, {\left (x e + d\right )}^{\frac {3}{2}} a d + 225 \, \sqrt {x e + d} a d^{2}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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