Optimal. Leaf size=480 \[ -\frac {2 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^6}+\frac {4 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}-\frac {5 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}+\frac {40 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {5 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}+\frac {4 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6} \]
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Rubi [A] time = 0.48, antiderivative size = 355, normalized size of antiderivative = 0.74, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 43
Rule 2301
Rule 2334
Rule 2398
Rule 2411
Rule 2454
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx &=2 \operatorname {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} (2 b e n) \operatorname {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} (2 b n) \operatorname {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt {x}\right )\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {1}{3} \left (2 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+e \sqrt {x}\right )\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+e \sqrt {x}\right )}{90 e^6}\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \operatorname {Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac {60 d^6 \log (x)}{x}\right ) \, dx,x,d+e \sqrt {x}\right )}{90 e^6}\\ &=\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (2 b^2 d^6 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt {x}\right )}{3 e^6}\\ &=\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}+\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.33, size = 295, normalized size = 0.61 \[ \frac {e \sqrt {x} \left (1800 a^2 e^5 x^{5/2}+60 a b n \left (60 d^5-30 d^4 e \sqrt {x}+20 d^3 e^2 x-15 d^2 e^3 x^{3/2}+12 d e^4 x^2-10 e^5 x^{5/2}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt {x}-1140 d^3 e^2 x+555 d^2 e^3 x^{3/2}-264 d e^4 x^2+100 e^5 x^{5/2}\right )\right )-60 b \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-147 d^6-60 d^5 e \sqrt {x}+30 d^4 e^2 x-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^3\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{5400 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 487, normalized size = 1.01 \[ \frac {1800 \, b^{2} e^{6} x^{3} \log \relax (c)^{2} + 100 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{3} + 15 \, {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x^{2} + 1800 \, {\left (b^{2} e^{6} n^{2} x^{3} - b^{2} d^{6} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} + 90 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x - 60 \, {\left (15 \, b^{2} d^{2} e^{4} n^{2} x^{2} + 30 \, b^{2} d^{4} e^{2} n^{2} x - 147 \, b^{2} d^{6} n^{2} + 60 \, a b d^{6} n + 10 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{3} - 60 \, {\left (b^{2} e^{6} n x^{3} - b^{2} d^{6} n\right )} \log \relax (c) - 4 \, {\left (3 \, b^{2} d e^{5} n^{2} x^{2} + 5 \, b^{2} d^{3} e^{3} n^{2} x + 15 \, b^{2} d^{5} e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) - 300 \, {\left (3 \, b^{2} d^{2} e^{4} n x^{2} + 6 \, b^{2} d^{4} e^{2} n x + 2 \, {\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{3}\right )} \log \relax (c) - 12 \, {\left (735 \, b^{2} d^{5} e n^{2} - 300 \, a b d^{5} e n + 2 \, {\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x^{2} + 5 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x - 20 \, {\left (3 \, b^{2} d e^{5} n x^{2} + 5 \, b^{2} d^{3} e^{3} n x + 15 \, b^{2} d^{5} e n\right )} \log \relax (c)\right )} \sqrt {x}}{5400 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 956, normalized size = 1.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{2} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 324, normalized size = 0.68 \[ \frac {1}{3} \, b^{2} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + \frac {2}{3} \, a b x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{2} x^{3} - \frac {1}{90} \, a b e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} - \frac {1}{5400} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{3} - 264 \, d e^{5} x^{\frac {5}{2}} + 555 \, d^{2} e^{4} x^{2} + 1800 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 1140 \, d^{3} e^{3} x^{\frac {3}{2}} + 2610 \, d^{4} e^{2} x + 8820 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 8820 \, d^{5} e \sqrt {x}\right )} n^{2}}{e^{6}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.78, size = 434, normalized size = 0.90 \[ \frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3}+\frac {b^2\,n^2\,x^3}{54}+\frac {2\,a\,b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3\,e^6}-\frac {a\,b\,n\,x^3}{9}-\frac {b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,\sqrt {x}\right )}{30\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^2}{360\,e^2}-\frac {19\,b^2\,d^3\,n^2\,x^{3/2}}{90\,e^3}-\frac {49\,b^2\,d^5\,n^2\,\sqrt {x}}{30\,e^5}-\frac {11\,b^2\,d\,n^2\,x^{5/2}}{225\,e}+\frac {29\,b^2\,d^4\,n^2\,x}{60\,e^4}-\frac {b^2\,d^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{6\,e^2}+\frac {2\,b^2\,d^3\,n\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9\,e^3}+\frac {2\,b^2\,d^5\,n\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^5}+\frac {2\,a\,b\,d\,n\,x^{5/2}}{15\,e}-\frac {a\,b\,d^4\,n\,x}{3\,e^4}-\frac {2\,a\,b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{3\,e^6}+\frac {2\,b^2\,d\,n\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{15\,e}-\frac {b^2\,d^4\,n\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^4}-\frac {a\,b\,d^2\,n\,x^2}{6\,e^2}+\frac {2\,a\,b\,d^3\,n\,x^{3/2}}{9\,e^3}+\frac {2\,a\,b\,d^5\,n\,\sqrt {x}}{3\,e^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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