Optimal. Leaf size=156 \[ -\frac {2 (b c-a d)^5 (c+d x)^{3/2}}{3 d^6}+\frac {2 b (b c-a d)^4 (c+d x)^{5/2}}{d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{9/2}}{9 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{11/2}}{11 d^6}+\frac {2 b^5 (c+d x)^{13/2}}{13 d^6} \]
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Rubi [A]
time = 0.04, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45}
\begin {gather*} -\frac {10 b^4 (c+d x)^{11/2} (b c-a d)}{11 d^6}+\frac {20 b^3 (c+d x)^{9/2} (b c-a d)^2}{9 d^6}-\frac {20 b^2 (c+d x)^{7/2} (b c-a d)^3}{7 d^6}+\frac {2 b (c+d x)^{5/2} (b c-a d)^4}{d^6}-\frac {2 (c+d x)^{3/2} (b c-a d)^5}{3 d^6}+\frac {2 b^5 (c+d x)^{13/2}}{13 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^5 \sqrt {c+d x} \, dx &=\int \left (\frac {(-b c+a d)^5 \sqrt {c+d x}}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^{3/2}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{5/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{7/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{9/2}}{d^5}+\frac {b^5 (c+d x)^{11/2}}{d^5}\right ) \, dx\\ &=-\frac {2 (b c-a d)^5 (c+d x)^{3/2}}{3 d^6}+\frac {2 b (b c-a d)^4 (c+d x)^{5/2}}{d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{9/2}}{9 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{11/2}}{11 d^6}+\frac {2 b^5 (c+d x)^{13/2}}{13 d^6}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 217, normalized size = 1.39 \begin {gather*} \frac {2 (c+d x)^{3/2} \left (3003 a^5 d^5+3003 a^4 b d^4 (-2 c+3 d x)+858 a^3 b^2 d^3 \left (8 c^2-12 c d x+15 d^2 x^2\right )+286 a^2 b^3 d^2 \left (-16 c^3+24 c^2 d x-30 c d^2 x^2+35 d^3 x^3\right )+13 a b^4 d \left (128 c^4-192 c^3 d x+240 c^2 d^2 x^2-280 c d^3 x^3+315 d^4 x^4\right )+b^5 \left (-256 c^5+384 c^4 d x-480 c^3 d^2 x^2+560 c^2 d^3 x^3-630 c d^4 x^4+693 d^5 x^5\right )\right )}{9009 d^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 7.07, size = 247, normalized size = 1.58 \begin {gather*} \frac {2 \left (3003 a^5 d^5+9009 b \left (a^4 d^4-4 a^3 b c d^3+6 a^2 b^2 c^2 d^2-4 a b^3 c^3 d+b^4 c^4\right ) \left (c+d x\right )-15015 a^4 b c d^4+12870 b^2 \left (a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right ) \left (c+d x\right )^2+30030 a^3 b^2 c^2 d^3+10010 b^3 \left (a^2 d^2-2 a b c d+b^2 c^2\right ) \left (c+d x\right )^3-30030 a^2 b^3 c^3 d^2+4095 b^4 \left (a d-b c\right ) \left (c+d x\right )^4+15015 a b^4 c^4 d+693 b^5 \left (c+d x\right )^5-3003 b^5 c^5\right ) \left (c+d x\right )^{\frac {3}{2}}}{9009 d^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 121, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {5}{2}}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\) | \(121\) |
default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {5}{2}}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{6}}\) | \(121\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (693 b^{5} x^{5} d^{5}+4095 a \,b^{4} d^{5} x^{4}-630 b^{5} c \,d^{4} x^{4}+10010 a^{2} b^{3} d^{5} x^{3}-3640 a \,b^{4} c \,d^{4} x^{3}+560 b^{5} c^{2} d^{3} x^{3}+12870 a^{3} b^{2} d^{5} x^{2}-8580 a^{2} b^{3} c \,d^{4} x^{2}+3120 a \,b^{4} c^{2} d^{3} x^{2}-480 b^{5} c^{3} d^{2} x^{2}+9009 a^{4} b \,d^{5} x -10296 a^{3} b^{2} c \,d^{4} x +6864 a^{2} b^{3} c^{2} d^{3} x -2496 a \,b^{4} c^{3} d^{2} x +384 b^{5} c^{4} d x +3003 a^{5} d^{5}-6006 a^{4} b c \,d^{4}+6864 a^{3} b^{2} c^{2} d^{3}-4576 a^{2} b^{3} c^{3} d^{2}+1664 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{9009 d^{6}}\) | \(273\) |
trager | \(\frac {2 \left (693 b^{5} d^{6} x^{6}+4095 a \,b^{4} d^{6} x^{5}+63 b^{5} c \,d^{5} x^{5}+10010 a^{2} b^{3} d^{6} x^{4}+455 a \,b^{4} c \,d^{5} x^{4}-70 b^{5} c^{2} d^{4} x^{4}+12870 a^{3} b^{2} d^{6} x^{3}+1430 a^{2} b^{3} c \,d^{5} x^{3}-520 a \,b^{4} c^{2} d^{4} x^{3}+80 b^{5} c^{3} d^{3} x^{3}+9009 a^{4} b \,d^{6} x^{2}+2574 a^{3} b^{2} c \,d^{5} x^{2}-1716 a^{2} b^{3} c^{2} d^{4} x^{2}+624 a \,b^{4} c^{3} d^{3} x^{2}-96 b^{5} c^{4} d^{2} x^{2}+3003 a^{5} d^{6} x +3003 a^{4} b c \,d^{5} x -3432 a^{3} b^{2} c^{2} d^{4} x +2288 a^{2} b^{3} c^{3} d^{3} x -832 a \,b^{4} c^{4} d^{2} x +128 b^{5} c^{5} d x +3003 a^{5} c \,d^{5}-6006 a^{4} b \,c^{2} d^{4}+6864 a^{3} b^{2} c^{3} d^{3}-4576 a^{2} b^{3} c^{4} d^{2}+1664 a \,b^{4} c^{5} d -256 b^{5} c^{6}\right ) \sqrt {d x +c}}{9009 d^{6}}\) | \(361\) |
risch | \(\frac {2 \left (693 b^{5} d^{6} x^{6}+4095 a \,b^{4} d^{6} x^{5}+63 b^{5} c \,d^{5} x^{5}+10010 a^{2} b^{3} d^{6} x^{4}+455 a \,b^{4} c \,d^{5} x^{4}-70 b^{5} c^{2} d^{4} x^{4}+12870 a^{3} b^{2} d^{6} x^{3}+1430 a^{2} b^{3} c \,d^{5} x^{3}-520 a \,b^{4} c^{2} d^{4} x^{3}+80 b^{5} c^{3} d^{3} x^{3}+9009 a^{4} b \,d^{6} x^{2}+2574 a^{3} b^{2} c \,d^{5} x^{2}-1716 a^{2} b^{3} c^{2} d^{4} x^{2}+624 a \,b^{4} c^{3} d^{3} x^{2}-96 b^{5} c^{4} d^{2} x^{2}+3003 a^{5} d^{6} x +3003 a^{4} b c \,d^{5} x -3432 a^{3} b^{2} c^{2} d^{4} x +2288 a^{2} b^{3} c^{3} d^{3} x -832 a \,b^{4} c^{4} d^{2} x +128 b^{5} c^{5} d x +3003 a^{5} c \,d^{5}-6006 a^{4} b \,c^{2} d^{4}+6864 a^{3} b^{2} c^{3} d^{3}-4576 a^{2} b^{3} c^{4} d^{2}+1664 a \,b^{4} c^{5} d -256 b^{5} c^{6}\right ) \sqrt {d x +c}}{9009 d^{6}}\) | \(361\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 259, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (693 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{5} - 4095 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 12870 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 3003 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{9009 \, d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 338 vs.
\(2 (134) = 268\).
time = 0.30, size = 338, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} d^{6} x^{6} - 256 \, b^{5} c^{6} + 1664 \, a b^{4} c^{5} d - 4576 \, a^{2} b^{3} c^{4} d^{2} + 6864 \, a^{3} b^{2} c^{3} d^{3} - 6006 \, a^{4} b c^{2} d^{4} + 3003 \, a^{5} c d^{5} + 63 \, {\left (b^{5} c d^{5} + 65 \, a b^{4} d^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{2} d^{4} - 13 \, a b^{4} c d^{5} - 286 \, a^{2} b^{3} d^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} c^{3} d^{3} - 52 \, a b^{4} c^{2} d^{4} + 143 \, a^{2} b^{3} c d^{5} + 1287 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{4} d^{2} - 208 \, a b^{4} c^{3} d^{3} + 572 \, a^{2} b^{3} c^{2} d^{4} - 858 \, a^{3} b^{2} c d^{5} - 3003 \, a^{4} b d^{6}\right )} x^{2} + {\left (128 \, b^{5} c^{5} d - 832 \, a b^{4} c^{4} d^{2} + 2288 \, a^{2} b^{3} c^{3} d^{3} - 3432 \, a^{3} b^{2} c^{2} d^{4} + 3003 \, a^{4} b c d^{5} + 3003 \, a^{5} d^{6}\right )} x\right )} \sqrt {d x + c}}{9009 \, d^{6}} \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. 314 vs.
\(2 (144) = 288\).
time = 2.41, size = 314, normalized size = 2.01 \begin {gather*} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{3 d^{5}}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 641 vs.
\(2 (134) = 268\).
time = 0.01, size = 1039, normalized size = 6.66 \begin {gather*} \frac {\frac {2 b^{5} d \left (\frac {1}{13} \sqrt {c+d x} \left (c+d x\right )^{6}-\frac {6}{11} \sqrt {c+d x} \left (c+d x\right )^{5} c+\frac {5}{3} \sqrt {c+d x} \left (c+d x\right )^{4} c^{2}-\frac {20}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c^{3}+3 \sqrt {c+d x} \left (c+d x\right )^{2} c^{4}-2 \sqrt {c+d x} \left (c+d x\right ) c^{5}+\sqrt {c+d x} c^{6}\right )}{d^{6}}+\frac {2 b^{5} c \left (\frac {1}{11} \sqrt {c+d x} \left (c+d x\right )^{5}-\frac {5}{9} \sqrt {c+d x} \left (c+d x\right )^{4} c+\frac {10}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c^{2}-2 \sqrt {c+d x} \left (c+d x\right )^{2} c^{3}+\frac {5}{3} \sqrt {c+d x} \left (c+d x\right ) c^{4}-\sqrt {c+d x} c^{5}\right )}{d^{5}}+\frac {10 a b^{4} d \left (\frac {1}{11} \sqrt {c+d x} \left (c+d x\right )^{5}-\frac {5}{9} \sqrt {c+d x} \left (c+d x\right )^{4} c+\frac {10}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c^{2}-2 \sqrt {c+d x} \left (c+d x\right )^{2} c^{3}+\frac {5}{3} \sqrt {c+d x} \left (c+d x\right ) c^{4}-\sqrt {c+d x} c^{5}\right )}{d^{5}}+\frac {10 a b^{4} c \left (\frac {1}{9} \sqrt {c+d x} \left (c+d x\right )^{4}-\frac {4}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c+\frac {6}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c^{2}-\frac {4}{3} \sqrt {c+d x} \left (c+d x\right ) c^{3}+\sqrt {c+d x} c^{4}\right )}{d^{4}}+\frac {20 a^{2} b^{3} d \left (\frac {1}{9} \sqrt {c+d x} \left (c+d x\right )^{4}-\frac {4}{7} \sqrt {c+d x} \left (c+d x\right )^{3} c+\frac {6}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c^{2}-\frac {4}{3} \sqrt {c+d x} \left (c+d x\right ) c^{3}+\sqrt {c+d x} c^{4}\right )}{d^{4}}+\frac {20 a^{2} b^{3} c \left (\frac {1}{7} \sqrt {c+d x} \left (c+d x\right )^{3}-\frac {3}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c+\sqrt {c+d x} \left (c+d x\right ) c^{2}-\sqrt {c+d x} c^{3}\right )}{d^{3}}+\frac {20 a^{3} b^{2} d \left (\frac {1}{7} \sqrt {c+d x} \left (c+d x\right )^{3}-\frac {3}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c+\sqrt {c+d x} \left (c+d x\right ) c^{2}-\sqrt {c+d x} c^{3}\right )}{d^{3}}+\frac {20 a^{3} b^{2} c \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+\frac {10 a^{4} b d \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+2 a^{5} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+\frac {10 a^{4} b c \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a^{5} c \sqrt {c+d x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 137, normalized size = 0.88 \begin {gather*} \frac {2\,b^5\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{5/2}}{d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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