Optimal. Leaf size=128 \[ -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78}
\begin {gather*} -\frac {2 b (d+e x)^{9/2} (-2 a B e-A b e+3 b B d)}{9 e^4}+\frac {2 (d+e x)^{7/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^{3/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{3/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^3}+\frac {b^2 B (d+e x)^{9/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{5/2}}{5 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{7/2}}{7 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{9/2}}{9 e^4}+\frac {2 b^2 B (d+e x)^{11/2}}{11 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 139, normalized size = 1.09 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (99 a^2 e^2 (-2 B d+7 A e+5 B e x)+22 a b e \left (9 A e (-2 d+5 e x)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+b^2 \left (11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )-3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 122, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(122\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (315 b^{2} B \,x^{3} e^{3}+385 A \,b^{2} e^{3} x^{2}+770 B a b \,e^{3} x^{2}-210 B \,b^{2} d \,e^{2} x^{2}+990 A a b \,e^{3} x -220 A \,b^{2} d \,e^{2} x +495 B \,a^{2} e^{3} x -440 B a b d \,e^{2} x +120 B \,b^{2} d^{2} e x +693 a^{2} A \,e^{3}-396 A a b d \,e^{2}+88 A \,b^{2} d^{2} e -198 B \,a^{2} d \,e^{2}+176 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{3465 e^{4}}\) | \(169\) |
trager | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
risch | \(\frac {2 \left (315 B \,b^{2} e^{5} x^{5}+385 A \,b^{2} e^{5} x^{4}+770 B a b \,e^{5} x^{4}+420 B \,b^{2} d \,e^{4} x^{4}+990 A a b \,e^{5} x^{3}+550 A \,b^{2} d \,e^{4} x^{3}+495 B \,a^{2} e^{5} x^{3}+1100 B a b d \,e^{4} x^{3}+15 B \,b^{2} d^{2} e^{3} x^{3}+693 A \,a^{2} e^{5} x^{2}+1584 A a b d \,e^{4} x^{2}+33 A \,b^{2} d^{2} e^{3} x^{2}+792 B \,a^{2} d \,e^{4} x^{2}+66 B a b \,d^{2} e^{3} x^{2}-18 B \,b^{2} d^{3} e^{2} x^{2}+1386 A \,a^{2} d \,e^{4} x +198 A a b \,d^{2} e^{3} x -44 A \,b^{2} d^{3} e^{2} x +99 B \,a^{2} d^{2} e^{3} x -88 B a b \,d^{3} e^{2} x +24 B \,b^{2} d^{4} e x +693 A \,a^{2} d^{2} e^{3}-396 A a b \,d^{3} e^{2}+88 A \,b^{2} d^{4} e -198 B \,a^{2} d^{3} e^{2}+176 B a b \,d^{4} e -48 B \,b^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{4}}\) | \(341\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 168, normalized size = 1.31 \begin {gather*} \frac {2}{3465} \, {\left (315 \, {\left (x e + d\right )}^{\frac {11}{2}} B b^{2} - 385 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 495 \, {\left (3 \, B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 693 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b e + A b^{2} e\right )} d^{2} + {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-4\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. 277 vs.
\(2 (119) = 238\).
time = 1.15, size = 277, normalized size = 2.16 \begin {gather*} -\frac {2}{3465} \, {\left (48 \, B b^{2} d^{5} - {\left (315 \, B b^{2} x^{5} + 693 \, A a^{2} x^{2} + 385 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 495 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} e^{5} - 2 \, {\left (210 \, B b^{2} d x^{4} + 693 \, A a^{2} d x + 275 \, {\left (2 \, B a b + A b^{2}\right )} d x^{3} + 396 \, {\left (B a^{2} + 2 \, A a b\right )} d x^{2}\right )} e^{4} - 3 \, {\left (5 \, B b^{2} d^{2} x^{3} + 231 \, A a^{2} d^{2} + 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x^{2} + 33 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} x\right )} e^{3} + 2 \, {\left (9 \, B b^{2} d^{3} x^{2} + 22 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} x + 99 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} e^{2} - 8 \, {\left (3 \, B b^{2} d^{4} x + 11 \, {\left (2 \, B a b + A b^{2}\right )} d^{4}\right )} e\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 13.21, size = 586, normalized size = 4.58 \begin {gather*} A a^{2} d \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {2 A a^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {4 A a b d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 A a b \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 A b^{2} d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {2 A b^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B a^{2} d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {2 B a^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {4 B a b d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {4 B a b \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {2 B b^{2} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {2 B b^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{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. 901 vs.
\(2 (119) = 238\).
time = 0.58, size = 901, normalized size = 7.04 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} d^{2} e^{\left (-1\right )} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a b d^{2} e^{\left (-1\right )} + 462 \, {\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 a b d^{2} e^{\left (-2\right )} + 231 \, {\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^{2} d^{2} e^{\left (-2\right )} + 99 \, {\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 b^{2} d^{2} e^{\left (-3\right )} + 462 \, {\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 a^{2} d e^{\left (-1\right )} + 924 \, {\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 a b d e^{\left (-1\right )} + 396 \, {\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 a b d e^{\left (-2\right )} + 198 \, {\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^{2} d e^{\left (-2\right )} + 22 \, {\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 b^{2} d e^{\left (-3\right )} + 3465 \, \sqrt {x e + d} A a^{2} d^{2} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2} d + 99 \, {\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 a^{2} e^{\left (-1\right )} + 198 \, {\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 a b e^{\left (-1\right )} + 22 \, {\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 a b e^{\left (-2\right )} + 11 \, {\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^{2} e^{\left (-2\right )} + 5 \, {\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 b^{2} e^{\left (-3\right )} + 231 \, {\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 a^{2}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{9\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{7\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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