Optimal. Leaf size=100 \[ -\frac {2 (b c-a d)^3 (c+d x)^{5/2}}{5 d^4}+\frac {6 b (b c-a d)^2 (c+d x)^{7/2}}{7 d^4}-\frac {2 b^2 (b c-a d) (c+d x)^{9/2}}{3 d^4}+\frac {2 b^3 (c+d x)^{11/2}}{11 d^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, 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 {2 b^2 (c+d x)^{9/2} (b c-a d)}{3 d^4}+\frac {6 b (c+d x)^{7/2} (b c-a d)^2}{7 d^4}-\frac {2 (c+d x)^{5/2} (b c-a d)^3}{5 d^4}+\frac {2 b^3 (c+d x)^{11/2}}{11 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d)^3 (c+d x)^{3/2}}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{5/2}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{7/2}}{d^3}+\frac {b^3 (c+d x)^{9/2}}{d^3}\right ) \, dx\\ &=-\frac {2 (b c-a d)^3 (c+d x)^{5/2}}{5 d^4}+\frac {6 b (b c-a d)^2 (c+d x)^{7/2}}{7 d^4}-\frac {2 b^2 (b c-a d) (c+d x)^{9/2}}{3 d^4}+\frac {2 b^3 (c+d x)^{11/2}}{11 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 102, normalized size = 1.02 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (231 a^3 d^3+99 a^2 b d^2 (-2 c+5 d x)+11 a b^2 d \left (8 c^2-20 c d x+35 d^2 x^2\right )+b^3 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )\right )}{1155 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 78, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {6 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{4}}\) | \(78\) |
default | \(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {6 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{4}}\) | \(78\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (105 b^{3} x^{3} d^{3}+385 a \,b^{2} d^{3} x^{2}-70 b^{3} c \,d^{2} x^{2}+495 a^{2} b \,d^{3} x -220 a \,b^{2} c \,d^{2} x +40 b^{3} c^{2} d x +231 a^{3} d^{3}-198 a^{2} b c \,d^{2}+88 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{1155 d^{4}}\) | \(116\) |
trager | \(\frac {2 \left (105 b^{3} d^{5} x^{5}+385 a \,b^{2} d^{5} x^{4}+140 b^{3} c \,d^{4} x^{4}+495 a^{2} b \,d^{5} x^{3}+550 a \,b^{2} c \,d^{4} x^{3}+5 b^{3} c^{2} d^{3} x^{3}+231 a^{3} d^{5} x^{2}+792 a^{2} b c \,d^{4} x^{2}+33 a \,b^{2} c^{2} d^{3} x^{2}-6 b^{3} c^{3} d^{2} x^{2}+462 a^{3} c \,d^{4} x +99 a^{2} b \,c^{2} d^{3} x -44 a \,b^{2} c^{3} d^{2} x +8 b^{3} c^{4} d x +231 a^{3} c^{2} d^{3}-198 a^{2} b \,c^{3} d^{2}+88 a \,b^{2} c^{4} d -16 b^{3} c^{5}\right ) \sqrt {d x +c}}{1155 d^{4}}\) | \(228\) |
risch | \(\frac {2 \left (105 b^{3} d^{5} x^{5}+385 a \,b^{2} d^{5} x^{4}+140 b^{3} c \,d^{4} x^{4}+495 a^{2} b \,d^{5} x^{3}+550 a \,b^{2} c \,d^{4} x^{3}+5 b^{3} c^{2} d^{3} x^{3}+231 a^{3} d^{5} x^{2}+792 a^{2} b c \,d^{4} x^{2}+33 a \,b^{2} c^{2} d^{3} x^{2}-6 b^{3} c^{3} d^{2} x^{2}+462 a^{3} c \,d^{4} x +99 a^{2} b \,c^{2} d^{3} x -44 a \,b^{2} c^{3} d^{2} x +8 b^{3} c^{4} d x +231 a^{3} c^{2} d^{3}-198 a^{2} b \,c^{3} d^{2}+88 a \,b^{2} c^{4} d -16 b^{3} c^{5}\right ) \sqrt {d x +c}}{1155 d^{4}}\) | \(228\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 118, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (105 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{3} - 385 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 495 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 231 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{1155 \, d^{4}} \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. 216 vs.
\(2 (84) = 168\).
time = 0.48, size = 216, normalized size = 2.16 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} d^{5} x^{5} - 16 \, b^{3} c^{5} + 88 \, a b^{2} c^{4} d - 198 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} + 35 \, {\left (4 \, b^{3} c d^{4} + 11 \, a b^{2} d^{5}\right )} x^{4} + 5 \, {\left (b^{3} c^{2} d^{3} + 110 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} c^{3} d^{2} - 11 \, a b^{2} c^{2} d^{3} - 264 \, a^{2} b c d^{4} - 77 \, a^{3} d^{5}\right )} x^{2} + {\left (8 \, b^{3} c^{4} d - 44 \, a b^{2} c^{3} d^{2} + 99 \, a^{2} b c^{2} d^{3} + 462 \, a^{3} c d^{4}\right )} x\right )} \sqrt {d x + c}}{1155 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.57, size = 386, normalized size = 3.86 \begin {gather*} a^{3} c \left (\begin {cases} \sqrt {c} x & \text {for}\: d = 0 \\\frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{3} \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d} + \frac {6 a^{2} b c \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} + \frac {6 a^{2} b \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{2}} + \frac {6 a b^{2} c \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} + \frac {6 a b^{2} \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} + \frac {2 b^{3} c \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} + \frac {2 b^{3} \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} \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. 566 vs.
\(2 (84) = 168\).
time = 1.46, size = 566, normalized size = 5.66 \begin {gather*} \frac {2 \, {\left (3465 \, \sqrt {d x + c} a^{3} c^{2} + 2310 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} c + \frac {3465 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b c^{2}}{d} + 231 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{3} + \frac {693 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2} c^{2}}{d^{2}} + \frac {1386 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{2} b c}{d} + \frac {99 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3} c^{2}}{d^{3}} + \frac {594 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b^{2} c}{d^{2}} + \frac {297 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} b}{d} + \frac {22 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{3} c}{d^{3}} + \frac {33 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{2}}{d^{2}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{3}}{d^{3}}\right )}}{3465 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (c+d\,x\right )}^{11/2}}{11\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {6\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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