Optimal. Leaf size=182 \[ \frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {6}{7} a \left (a b B+A \left (b^2+a c\right )\right ) x^{7/2}+\frac {2}{9} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{9/2}+\frac {2}{11} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11/2}+\frac {6}{13} c \left (b^2 B+A b c+a B c\right ) x^{13/2}+\frac {2}{15} c^2 (3 b B+A c) x^{15/2}+\frac {2}{17} B c^3 x^{17/2} \]
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Rubi [A]
time = 0.07, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {779}
\begin {gather*} \frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 x^{5/2} (a B+3 A b)+\frac {6}{13} c x^{13/2} \left (a B c+A b c+b^2 B\right )+\frac {6}{7} a x^{7/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{11} x^{11/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{9} x^{9/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{15} c^2 x^{15/2} (A c+3 b B)+\frac {2}{17} B c^3 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 779
Rubi steps
\begin {align*} \int \sqrt {x} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A \sqrt {x}+a^2 (3 A b+a B) x^{3/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{5/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{7/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{9/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{11/2}+c^2 (3 b B+A c) x^{13/2}+B c^3 x^{15/2}\right ) \, dx\\ &=\frac {2}{3} a^3 A x^{3/2}+\frac {2}{5} a^2 (3 A b+a B) x^{5/2}+\frac {6}{7} a \left (a b B+A \left (b^2+a c\right )\right ) x^{7/2}+\frac {2}{9} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{9/2}+\frac {2}{11} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11/2}+\frac {6}{13} c \left (b^2 B+A b c+a B c\right ) x^{13/2}+\frac {2}{15} c^2 (3 b B+A c) x^{15/2}+\frac {2}{17} B c^3 x^{17/2}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 178, normalized size = 0.98 \begin {gather*} \frac {2 x^{3/2} \left (51051 a^3 (5 A+3 B x)+7293 a^2 x (9 A (7 b+5 c x)+5 B x (9 b+7 c x))+255 a x^2 \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )+7 x^3 \left (17 A \left (715 b^3+1755 b^2 c x+1485 b c^2 x^2+429 c^3 x^3\right )+9 B x \left (1105 b^3+2805 b^2 c x+2431 b c^2 x^2+715 c^3 x^3\right )\right )\right )}{765765} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.85, size = 226, normalized size = 1.24
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (45045 B \,c^{3} x^{7}+51051 A \,c^{3} x^{6}+153153 B b \,c^{2} x^{6}+176715 A b \,c^{2} x^{5}+176715 a B \,c^{2} x^{5}+176715 B \,b^{2} c \,x^{5}+208845 a A \,c^{2} x^{4}+208845 A \,b^{2} c \,x^{4}+417690 B a b c \,x^{4}+69615 B \,b^{3} x^{4}+510510 A a b c \,x^{3}+85085 A \,b^{3} x^{3}+255255 a^{2} B c \,x^{3}+255255 B a \,b^{2} x^{3}+328185 a^{2} A c \,x^{2}+328185 A a \,b^{2} x^{2}+328185 a^{2} b B \,x^{2}+459459 A \,a^{2} b x +153153 B \,a^{3} x +255255 A \,a^{3}\right )}{765765}\) | \(192\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (45045 B \,c^{3} x^{7}+51051 A \,c^{3} x^{6}+153153 B b \,c^{2} x^{6}+176715 A b \,c^{2} x^{5}+176715 a B \,c^{2} x^{5}+176715 B \,b^{2} c \,x^{5}+208845 a A \,c^{2} x^{4}+208845 A \,b^{2} c \,x^{4}+417690 B a b c \,x^{4}+69615 B \,b^{3} x^{4}+510510 A a b c \,x^{3}+85085 A \,b^{3} x^{3}+255255 a^{2} B c \,x^{3}+255255 B a \,b^{2} x^{3}+328185 a^{2} A c \,x^{2}+328185 A a \,b^{2} x^{2}+328185 a^{2} b B \,x^{2}+459459 A \,a^{2} b x +153153 B \,a^{3} x +255255 A \,a^{3}\right )}{765765}\) | \(192\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (45045 B \,c^{3} x^{7}+51051 A \,c^{3} x^{6}+153153 B b \,c^{2} x^{6}+176715 A b \,c^{2} x^{5}+176715 a B \,c^{2} x^{5}+176715 B \,b^{2} c \,x^{5}+208845 a A \,c^{2} x^{4}+208845 A \,b^{2} c \,x^{4}+417690 B a b c \,x^{4}+69615 B \,b^{3} x^{4}+510510 A a b c \,x^{3}+85085 A \,b^{3} x^{3}+255255 a^{2} B c \,x^{3}+255255 B a \,b^{2} x^{3}+328185 a^{2} A c \,x^{2}+328185 A a \,b^{2} x^{2}+328185 a^{2} b B \,x^{2}+459459 A \,a^{2} b x +153153 B \,a^{3} x +255255 A \,a^{3}\right )}{765765}\) | \(192\) |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 A b \,c^{2}+B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{3} A \,x^{\frac {3}{2}}}{3}\) | \(226\) |
default | \(\frac {2 B \,c^{3} x^{\frac {17}{2}}}{17}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 A b \,c^{2}+B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {5}{2}}}{5}+\frac {2 a^{3} A \,x^{\frac {3}{2}}}{3}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 166, normalized size = 0.91 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {15}{2}} + \frac {6}{13} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {11}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 169, normalized size = 0.93 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B c^{3} x^{8} + 51051 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + 176715 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{6} + 69615 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + 255255 \, A a^{3} x + 85085 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + 328185 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + 153153 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.64, size = 216, normalized size = 1.19 \begin {gather*} \frac {2 A a^{3} x^{\frac {3}{2}}}{3} + \frac {2 B c^{3} x^{\frac {17}{2}}}{17} + \frac {2 x^{\frac {15}{2}} \left (A c^{3} + 3 B b c^{2}\right )}{15} + \frac {2 x^{\frac {13}{2}} \cdot \left (3 A b c^{2} + 3 B a c^{2} + 3 B b^{2} c\right )}{13} + \frac {2 x^{\frac {11}{2}} \cdot \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right )}{11} + \frac {2 x^{\frac {9}{2}} \cdot \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \cdot \left (3 A a^{2} c + 3 A a b^{2} + 3 B a^{2} b\right )}{7} + \frac {2 x^{\frac {5}{2}} \cdot \left (3 A a^{2} b + B a^{3}\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 193, normalized size = 1.06 \begin {gather*} \frac {2}{17} \, B c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b c^{2} x^{\frac {15}{2}} + \frac {2}{15} \, A c^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, B a c^{2} x^{\frac {13}{2}} + \frac {6}{13} \, A b c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {12}{11} \, B a b c x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {6}{11} \, A a c^{2} x^{\frac {11}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{3} \, B a^{2} c x^{\frac {9}{2}} + \frac {4}{3} \, A a b c x^{\frac {9}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} c x^{\frac {7}{2}} + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 169, normalized size = 0.93 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,c\,a^2}{3}+\frac {2\,B\,a\,b^2}{3}+\frac {4\,A\,c\,a\,b}{3}+\frac {2\,A\,b^3}{9}\right )+x^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,b^2\,c}{11}+\frac {12\,B\,a\,b\,c}{11}+\frac {6\,A\,a\,c^2}{11}\right )+x^{5/2}\,\left (\frac {2\,B\,a^3}{5}+\frac {6\,A\,b\,a^2}{5}\right )+x^{15/2}\,\left (\frac {2\,A\,c^3}{15}+\frac {2\,B\,b\,c^2}{5}\right )+x^{7/2}\,\left (\frac {6\,B\,a^2\,b}{7}+\frac {6\,A\,c\,a^2}{7}+\frac {6\,A\,a\,b^2}{7}\right )+x^{13/2}\,\left (\frac {6\,B\,b^2\,c}{13}+\frac {6\,A\,b\,c^2}{13}+\frac {6\,B\,a\,c^2}{13}\right )+\frac {2\,A\,a^3\,x^{3/2}}{3}+\frac {2\,B\,c^3\,x^{17/2}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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