Optimal. Leaf size=147 \[ \frac {2 d^2 (c d-b e)^2 (d+e x)^{9/2}}{9 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{15/2}}{15 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5} \]
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Rubi [A]
time = 0.05, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {712}
\begin {gather*} \frac {2 (d+e x)^{13/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{13 e^5}+\frac {2 d^2 (d+e x)^{9/2} (c d-b e)^2}{9 e^5}-\frac {4 c (d+e x)^{15/2} (2 c d-b e)}{15 e^5}-\frac {4 d (d+e x)^{11/2} (c d-b e) (2 c d-b e)}{11 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^{7/2}}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{9/2}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{13/2}}{e^4}+\frac {c^2 (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac {2 d^2 (c d-b e)^2 (d+e x)^{9/2}}{9 e^5}-\frac {4 d (c d-b e) (2 c d-b e) (d+e x)^{11/2}}{11 e^5}+\frac {2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{15/2}}{15 e^5}+\frac {2 c^2 (d+e x)^{17/2}}{17 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 124, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 144, normalized size = 0.98
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 c^{2} x^{4} e^{4}+14586 b c \,e^{4} x^{3}-3432 c^{2} d \,e^{3} x^{3}+8415 b^{2} e^{4} x^{2}-6732 b c d \,e^{3} x^{2}+1584 c^{2} d^{2} e^{2} x^{2}-3060 b^{2} d \,e^{3} x +2448 b c \,d^{2} e^{2} x -576 c^{2} d^{3} e x +680 d^{2} e^{2} b^{2}-544 b c \,d^{3} e +128 c^{2} d^{4}\right )}{109395 e^{5}}\) | \(141\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(144\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (-2 c^{2} d +2 \left (b e -c d \right ) c \right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (d^{2} c^{2}-4 d \left (b e -c d \right ) c +\left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 d^{2} \left (b e -c d \right ) c -2 d \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\) | \(144\) |
trager | \(\frac {2 \left (6435 c^{2} e^{8} x^{8}+14586 b c \,e^{8} x^{7}+22308 c^{2} d \,e^{7} x^{7}+8415 b^{2} e^{8} x^{6}+51612 b c d \,e^{7} x^{6}+26466 c^{2} d^{2} e^{6} x^{6}+30600 b^{2} d \,e^{7} x^{5}+63036 b c \,d^{2} e^{6} x^{5}+10908 c^{2} d^{3} e^{5} x^{5}+38930 b^{2} d^{2} e^{6} x^{4}+27200 b c \,d^{3} e^{5} x^{4}+35 c^{2} d^{4} e^{4} x^{4}+18020 b^{2} d^{3} e^{5} x^{3}+170 b c \,d^{4} e^{4} x^{3}-40 c^{2} d^{5} e^{3} x^{3}+255 b^{2} d^{4} e^{4} x^{2}-204 b c \,d^{5} e^{3} x^{2}+48 c^{2} d^{6} e^{2} x^{2}-340 b^{2} d^{5} e^{3} x +272 b c \,d^{6} e^{2} x -64 c^{2} d^{7} e x +680 b^{2} d^{6} e^{2}-544 b c \,d^{7} e +128 c^{2} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(305\) |
risch | \(\frac {2 \left (6435 c^{2} e^{8} x^{8}+14586 b c \,e^{8} x^{7}+22308 c^{2} d \,e^{7} x^{7}+8415 b^{2} e^{8} x^{6}+51612 b c d \,e^{7} x^{6}+26466 c^{2} d^{2} e^{6} x^{6}+30600 b^{2} d \,e^{7} x^{5}+63036 b c \,d^{2} e^{6} x^{5}+10908 c^{2} d^{3} e^{5} x^{5}+38930 b^{2} d^{2} e^{6} x^{4}+27200 b c \,d^{3} e^{5} x^{4}+35 c^{2} d^{4} e^{4} x^{4}+18020 b^{2} d^{3} e^{5} x^{3}+170 b c \,d^{4} e^{4} x^{3}-40 c^{2} d^{5} e^{3} x^{3}+255 b^{2} d^{4} e^{4} x^{2}-204 b c \,d^{5} e^{3} x^{2}+48 c^{2} d^{6} e^{2} x^{2}-340 b^{2} d^{5} e^{3} x +272 b c \,d^{6} e^{2} x -64 c^{2} d^{7} e x +680 b^{2} d^{6} e^{2}-544 b c \,d^{7} e +128 c^{2} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\) | \(305\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 144, normalized size = 0.98 \begin {gather*} \frac {2}{109395} \, {\left (6435 \, {\left (x e + d\right )}^{\frac {17}{2}} c^{2} - 14586 \, {\left (2 \, c^{2} d - b c e\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 8415 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 19890 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 12155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-5\right )} \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. 279 vs.
\(2 (131) = 262\).
time = 2.14, size = 279, normalized size = 1.90 \begin {gather*} \frac {2}{109395} \, {\left (128 \, c^{2} d^{8} + 33 \, {\left (195 \, c^{2} x^{8} + 442 \, b c x^{7} + 255 \, b^{2} x^{6}\right )} e^{8} + 12 \, {\left (1859 \, c^{2} d x^{7} + 4301 \, b c d x^{6} + 2550 \, b^{2} d x^{5}\right )} e^{7} + 2 \, {\left (13233 \, c^{2} d^{2} x^{6} + 31518 \, b c d^{2} x^{5} + 19465 \, b^{2} d^{2} x^{4}\right )} e^{6} + 4 \, {\left (2727 \, c^{2} d^{3} x^{5} + 6800 \, b c d^{3} x^{4} + 4505 \, b^{2} d^{3} x^{3}\right )} e^{5} + 5 \, {\left (7 \, c^{2} d^{4} x^{4} + 34 \, b c d^{4} x^{3} + 51 \, b^{2} d^{4} x^{2}\right )} e^{4} - 4 \, {\left (10 \, c^{2} d^{5} x^{3} + 51 \, b c d^{5} x^{2} + 85 \, b^{2} d^{5} x\right )} e^{3} + 8 \, {\left (6 \, c^{2} d^{6} x^{2} + 34 \, b c d^{6} x + 85 \, b^{2} d^{6}\right )} e^{2} - 32 \, {\left (2 \, c^{2} d^{7} x + 17 \, b c d^{7}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \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. 590 vs.
\(2 (144) = 288\).
time = 0.73, size = 590, normalized size = 4.01 \begin {gather*} \begin {cases} \frac {16 b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {64 b c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {32 b c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {8 b c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {4 b c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {640 b c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {824 b c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {184 b c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {4 b c e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 c^{2} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 c^{2} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 c^{2} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 c^{2} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 c^{2} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 c^{2} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 c^{2} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 c^{2} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 c^{2} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {b^{2} x^{3}}{3} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5}\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. 1245 vs.
\(2 (131) = 262\).
time = 1.29, size = 1245, normalized size = 8.47 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 138, normalized size = 0.94 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}-\frac {{\left (d+e\,x\right )}^{11/2}\,\left (4\,b^2\,d\,e^2-12\,b\,c\,d^2\,e+8\,c^2\,d^3\right )}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2\right )}{13\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,d^2\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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