Optimal. Leaf size=71 \[ \frac {2 (b c-a d)^2 (c+d x)^{5/2}}{5 d^3}-\frac {4 b (b c-a d) (c+d x)^{7/2}}{7 d^3}+\frac {2 b^2 (c+d x)^{9/2}}{9 d^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, 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 {4 b (c+d x)^{7/2} (b c-a d)}{7 d^3}+\frac {2 (c+d x)^{5/2} (b c-a d)^2}{5 d^3}+\frac {2 b^2 (c+d x)^{9/2}}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^{3/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{5/2}}{d^2}+\frac {b^2 (c+d x)^{7/2}}{d^2}\right ) \, dx\\ &=\frac {2 (b c-a d)^2 (c+d x)^{5/2}}{5 d^3}-\frac {4 b (b c-a d) (c+d x)^{7/2}}{7 d^3}+\frac {2 b^2 (c+d x)^{9/2}}{9 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (-2 c+5 d x)+b^2 \left (8 c^2-20 c d x+35 d^2 x^2\right )\right )}{315 d^3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(71)=142\).
time = 8.95, size = 142, normalized size = 2.00 \begin {gather*} \frac {2 \left (21 a^2 d^2 \left (-2 c+3 d x\right )+105 a^2 c d^2+6 a b d \left (-42 c \left (c+d x\right )-7 c \left (2 c-3 d x\right )+35 c^2+15 \left (c+d x\right )^2\right )+b^2 \left (-135 c \left (c+d x\right )^2+3 c \left (-42 c \left (c+d x\right )+35 c^2+15 \left (c+d x\right )^2\right )+189 c^2 \left (c+d x\right )-105 c^3+35 \left (c+d x\right )^3\right )\right ) \left (c+d x\right )^{\frac {3}{2}}}{315 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 56, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{3}}\) | \(56\) |
default | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{3}}\) | \(56\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (35 b^{2} x^{2} d^{2}+90 a b \,d^{2} x -20 b^{2} c d x +63 a^{2} d^{2}-36 a b c d +8 b^{2} c^{2}\right )}{315 d^{3}}\) | \(63\) |
trager | \(\frac {2 \left (35 b^{2} d^{4} x^{4}+90 a b \,d^{4} x^{3}+50 b^{2} c \,d^{3} x^{3}+63 a^{2} d^{4} x^{2}+144 a b c \,d^{3} x^{2}+3 b^{2} c^{2} d^{2} x^{2}+126 a^{2} c \,d^{3} x +18 a b \,c^{2} d^{2} x -4 b^{2} c^{3} d x +63 a^{2} c^{2} d^{2}-36 a b \,c^{3} d +8 b^{2} c^{4}\right ) \sqrt {d x +c}}{315 d^{3}}\) | \(141\) |
risch | \(\frac {2 \left (35 b^{2} d^{4} x^{4}+90 a b \,d^{4} x^{3}+50 b^{2} c \,d^{3} x^{3}+63 a^{2} d^{4} x^{2}+144 a b c \,d^{3} x^{2}+3 b^{2} c^{2} d^{2} x^{2}+126 a^{2} c \,d^{3} x +18 a b \,c^{2} d^{2} x -4 b^{2} c^{3} d x +63 a^{2} c^{2} d^{2}-36 a b \,c^{3} d +8 b^{2} c^{4}\right ) \sqrt {d x +c}}{315 d^{3}}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 68, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{2} - 90 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 63 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{315 \, d^{3}} \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. 137 vs.
\(2 (59) = 118\).
time = 0.31, size = 137, normalized size = 1.93 \begin {gather*} \frac {2 \, {\left (35 \, b^{2} d^{4} x^{4} + 8 \, b^{2} c^{4} - 36 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} + 10 \, {\left (5 \, b^{2} c d^{3} + 9 \, a b d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{2} d^{2} + 48 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{3} d - 9 \, a b c^{2} d^{2} - 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x + c}}{315 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.86, size = 240, normalized size = 3.38 \begin {gather*} a^{2} 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^{2} \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 {4 a 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 {4 a 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 {2 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 {2 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}} \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. 360 vs.
\(2 (59) = 118\).
time = 0.00, size = 571, normalized size = 8.04 \begin {gather*} \frac {\frac {2 b^{2} d^{2} \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 {4 a b d^{2} \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 {4 b^{2} c 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 {2 a^{2} d^{2} \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 {2 b^{2} c^{2} \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 {8 a b c 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}}+4 a^{2} c \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+\frac {4 a b c^{2} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a^{2} c^{2} \sqrt {c+d x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 68, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{5/2}\,\left (35\,b^2\,{\left (c+d\,x\right )}^2+63\,a^2\,d^2+63\,b^2\,c^2-90\,b^2\,c\,\left (c+d\,x\right )+90\,a\,b\,d\,\left (c+d\,x\right )-126\,a\,b\,c\,d\right )}{315\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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