Optimal. Leaf size=71 \[ \frac {2 (b d-a e)^2 (d+e x)^{9/2}}{9 e^3}-\frac {4 b (b d-a e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45}
\begin {gather*} -\frac {4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^{7/2}}{e^2}-\frac {2 b (b d-a e) (d+e x)^{9/2}}{e^2}+\frac {b^2 (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e)^2 (d+e x)^{9/2}}{9 e^3}-\frac {4 b (b d-a e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (-2 d+9 e x)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.54, size = 70, normalized size = 0.99
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} x^{2} e^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 a^{2} e^{2}-52 a b d e +8 b^{2} d^{2}\right )}{1287 e^{3}}\) | \(63\) |
derivativedivides | \(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\) | \(70\) |
default | \(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\) | \(70\) |
trager | \(\frac {2 \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 e^{6} a^{2} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\) | \(223\) |
risch | \(\frac {2 \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 e^{6} a^{2} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 71, normalized size = 1.00 \begin {gather*} \frac {2}{1287} \, {\left (99 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{2} - 234 \, {\left (b^{2} d - a b e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 143 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (61) = 122\).
time = 2.63, size = 202, normalized size = 2.85 \begin {gather*} \frac {2}{1287} \, {\left (8 \, b^{2} d^{6} + {\left (99 \, b^{2} x^{6} + 234 \, a b x^{5} + 143 \, a^{2} x^{4}\right )} e^{6} + 4 \, {\left (90 \, b^{2} d x^{5} + 221 \, a b d x^{4} + 143 \, a^{2} d x^{3}\right )} e^{5} + 2 \, {\left (229 \, b^{2} d^{2} x^{4} + 598 \, a b d^{2} x^{3} + 429 \, a^{2} d^{2} x^{2}\right )} e^{4} + 4 \, {\left (53 \, b^{2} d^{3} x^{3} + 156 \, a b d^{3} x^{2} + 143 \, a^{2} d^{3} x\right )} e^{3} + {\left (3 \, b^{2} d^{4} x^{2} + 26 \, a b d^{4} x + 143 \, a^{2} d^{4}\right )} e^{2} - 4 \, {\left (b^{2} d^{5} x + 13 \, a b d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs.
\(2 (65) = 130\).
time = 0.63, size = 432, normalized size = 6.08 \begin {gather*} \begin {cases} \frac {2 a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 a b e^{3} x^{5} \sqrt {d + e x}}{11} + \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} & \text {for}\: e \neq 0 \\d^{\frac {7}{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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 840 vs.
\(2 (61) = 122\).
time = 1.02, size = 840, normalized size = 11.83 \begin {gather*} \frac {2}{45045} \, {\left (30030 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b d^{4} e^{\left (-1\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b^{2} d^{4} e^{\left (-2\right )} + 24024 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b d^{3} e^{\left (-1\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{2} d^{3} e^{\left (-2\right )} + 45045 \, \sqrt {x e + d} a^{2} d^{4} + 60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d^{3} + 15444 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b d^{2} e^{\left (-1\right )} + 858 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{2} d^{2} e^{\left (-2\right )} + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} d^{2} + 1144 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b d e^{\left (-1\right )} + 260 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{2} d e^{\left (-2\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} d + 130 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b e^{\left (-1\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{2} e^{\left (-2\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 68, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,b^2\,{\left (d+e\,x\right )}^2+143\,a^2\,e^2+143\,b^2\,d^2-234\,b^2\,d\,\left (d+e\,x\right )+234\,a\,b\,e\,\left (d+e\,x\right )-286\,a\,b\,d\,e\right )}{1287\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________