Optimal. Leaf size=71 \[ \frac {2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^3}-\frac {4 b (b c-a d) (c+d x)^{9/2}}{9 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^3} \]
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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)^{9/2} (b c-a d)}{9 d^3}+\frac {2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 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)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^2 (c+d x)^{5/2}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{7/2}}{d^2}+\frac {b^2 (c+d x)^{9/2}}{d^2}\right ) \, dx\\ &=\frac {2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^3}-\frac {4 b (b c-a d) (c+d x)^{9/2}}{9 d^3}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (-2 c+7 d x)+b^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{693 d^3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.64, size = 183, normalized size = 2.58 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (8 b^2 c^5-4 b c^4 d \left (11 a+b x\right )+c^3 d^2 \left (99 a^2+22 a b x+3 b^2 x^2\right )+d^3 x \left (297 a^2 c^2+297 a^2 c d x+99 a^2 d^2 x^2+330 a b c^2 x+418 a b c d x^2+154 a b d^2 x^3+113 b^2 c^2 x^2+161 b^2 c d x^3+63 b^2 d^2 x^4\right )\right ) \sqrt {c+d x}}{693 d^3},d\text {!=}0\right \}\right \},c^{\frac {5}{2}} \left (a^2 x+a b x^2+\frac {b^2 x^3}{3}\right )\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.15, size = 56, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{3}}\) | \(56\) |
default | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{3}}\) | \(56\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (63 b^{2} x^{2} d^{2}+154 a b \,d^{2} x -28 b^{2} c d x +99 a^{2} d^{2}-44 a b c d +8 b^{2} c^{2}\right )}{693 d^{3}}\) | \(63\) |
trager | \(\frac {2 \left (63 b^{2} d^{5} x^{5}+154 a b \,d^{5} x^{4}+161 b^{2} c \,d^{4} x^{4}+99 a^{2} d^{5} x^{3}+418 a b c \,d^{4} x^{3}+113 b^{2} c^{2} d^{3} x^{3}+297 a^{2} c \,d^{4} x^{2}+330 a b \,c^{2} d^{3} x^{2}+3 b^{2} c^{3} d^{2} x^{2}+297 a^{2} c^{2} d^{3} x +22 a b \,c^{3} d^{2} x -4 b^{2} c^{4} d x +99 a^{2} c^{3} d^{2}-44 a b \,c^{4} d +8 b^{2} c^{5}\right ) \sqrt {d x +c}}{693 d^{3}}\) | \(182\) |
risch | \(\frac {2 \left (63 b^{2} d^{5} x^{5}+154 a b \,d^{5} x^{4}+161 b^{2} c \,d^{4} x^{4}+99 a^{2} d^{5} x^{3}+418 a b c \,d^{4} x^{3}+113 b^{2} c^{2} d^{3} x^{3}+297 a^{2} c \,d^{4} x^{2}+330 a b \,c^{2} d^{3} x^{2}+3 b^{2} c^{3} d^{2} x^{2}+297 a^{2} c^{2} d^{3} x +22 a b \,c^{3} d^{2} x -4 b^{2} c^{4} d x +99 a^{2} c^{3} d^{2}-44 a b \,c^{4} d +8 b^{2} c^{5}\right ) \sqrt {d x +c}}{693 d^{3}}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 68, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{2} - 154 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 99 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{693 \, 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. 174 vs.
\(2 (59) = 118\).
time = 0.29, size = 174, normalized size = 2.45 \begin {gather*} \frac {2 \, {\left (63 \, b^{2} d^{5} x^{5} + 8 \, b^{2} c^{5} - 44 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} + 7 \, {\left (23 \, b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{4} + {\left (113 \, b^{2} c^{2} d^{3} + 418 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} + 3 \, {\left (b^{2} c^{3} d^{2} + 110 \, a b c^{2} d^{3} + 99 \, a^{2} c d^{4}\right )} x^{2} - {\left (4 \, b^{2} c^{4} d - 22 \, a b c^{3} d^{2} - 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{693 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.38, size = 355, normalized size = 5.00 \begin {gather*} \begin {cases} \frac {2 a^{2} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{2} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{2} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{2} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {8 a b c^{4} \sqrt {c + d x}}{63 d^{2}} + \frac {4 a b c^{3} x \sqrt {c + d x}}{63 d} + \frac {20 a b c^{2} x^{2} \sqrt {c + d x}}{21} + \frac {76 a b c d x^{3} \sqrt {c + d x}}{63} + \frac {4 a b d^{2} x^{4} \sqrt {c + d x}}{9} + \frac {16 b^{2} c^{5} \sqrt {c + d x}}{693 d^{3}} - \frac {8 b^{2} c^{4} x \sqrt {c + d x}}{693 d^{2}} + \frac {2 b^{2} c^{3} x^{2} \sqrt {c + d x}}{231 d} + \frac {226 b^{2} c^{2} x^{3} \sqrt {c + d x}}{693} + \frac {46 b^{2} c d x^{4} \sqrt {c + d x}}{99} + \frac {2 b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \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. 558 vs.
\(2 (59) = 118\).
time = 0.01, size = 911, normalized size = 12.83 \begin {gather*} \frac {\frac {2 b^{2} d^{3} \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 {4 a b d^{3} \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 {6 b^{2} c 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 {2 a^{2} d^{3} \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 {6 b^{2} c^{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 {12 a b c 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 {2 b^{2} c^{3} \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 {6 a^{2} c 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 {12 a b c^{2} 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}}+6 a^{2} c^{2} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+\frac {4 a b c^{3} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a^{2} c^{3} \sqrt {c+d x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 68, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}\,\left (63\,b^2\,{\left (c+d\,x\right )}^2+99\,a^2\,d^2+99\,b^2\,c^2-154\,b^2\,c\,\left (c+d\,x\right )+154\,a\,b\,d\,\left (c+d\,x\right )-198\,a\,b\,c\,d\right )}{693\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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