Optimal. Leaf size=83 \[ \frac {2 (b d-a e) (B d-A e) (d+e x)^{9/2}}{9 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b B (d+e x)^{13/2}}{13 e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78}
\begin {gather*} -\frac {2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac {2 b B (d+e x)^{13/2}}{13 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^{7/2} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^{7/2}}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^2}+\frac {b B (d+e x)^{11/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) (d+e x)^{9/2}}{9 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b B (d+e x)^{13/2}}{13 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 70, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (13 A b e (-2 d+9 e x)+13 a e (-2 B d+11 A e+9 B e x)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 73, normalized size = 0.88
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b B \,x^{2} e^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right )}{1287 e^{3}}\) | \(73\) |
derivativedivides | \(\frac {\frac {2 B b \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\) | \(73\) |
default | \(\frac {\frac {2 B b \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) B +b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (A e -B d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\) | \(73\) |
trager | \(\frac {2 \left (99 B b \,e^{6} x^{6}+117 A b \,e^{6} x^{5}+117 B a \,e^{6} x^{5}+360 B b d \,e^{5} x^{5}+143 A a \,e^{6} x^{4}+442 A b d \,e^{5} x^{4}+442 B a d \,e^{5} x^{4}+458 B b \,d^{2} e^{4} x^{4}+572 A a d \,e^{5} x^{3}+598 A b \,d^{2} e^{4} x^{3}+598 B a \,d^{2} e^{4} x^{3}+212 B b \,d^{3} e^{3} x^{3}+858 A a \,d^{2} e^{4} x^{2}+312 A b \,d^{3} e^{3} x^{2}+312 B a \,d^{3} e^{3} x^{2}+3 B b \,d^{4} e^{2} x^{2}+572 A a \,d^{3} e^{3} x +13 A b \,d^{4} e^{2} x +13 B a \,d^{4} e^{2} x -4 B b \,d^{5} e x +143 A a \,d^{4} e^{2}-26 A b \,d^{5} e -26 B a \,d^{5} e +8 B b \,d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\) | \(277\) |
risch | \(\frac {2 \left (99 B b \,e^{6} x^{6}+117 A b \,e^{6} x^{5}+117 B a \,e^{6} x^{5}+360 B b d \,e^{5} x^{5}+143 A a \,e^{6} x^{4}+442 A b d \,e^{5} x^{4}+442 B a d \,e^{5} x^{4}+458 B b \,d^{2} e^{4} x^{4}+572 A a d \,e^{5} x^{3}+598 A b \,d^{2} e^{4} x^{3}+598 B a \,d^{2} e^{4} x^{3}+212 B b \,d^{3} e^{3} x^{3}+858 A a \,d^{2} e^{4} x^{2}+312 A b \,d^{3} e^{3} x^{2}+312 B a \,d^{3} e^{3} x^{2}+3 B b \,d^{4} e^{2} x^{2}+572 A a \,d^{3} e^{3} x +13 A b \,d^{4} e^{2} x +13 B a \,d^{4} e^{2} x -4 B b \,d^{5} e x +143 A a \,d^{4} e^{2}-26 A b \,d^{5} e -26 B a \,d^{5} e +8 B b \,d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\) | \(277\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 81, normalized size = 0.98 \begin {gather*} \frac {2}{1287} \, {\left (99 \, {\left (x e + d\right )}^{\frac {13}{2}} B b - 117 \, {\left (2 \, B b d - B a e - A b e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 143 \, {\left (B b d^{2} + A a e^{2} - {\left (B a e + A b e\right )} d\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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs.
\(2 (75) = 150\).
time = 1.28, size = 221, normalized size = 2.66 \begin {gather*} \frac {2}{1287} \, {\left (8 \, B b d^{6} + {\left (99 \, B b x^{6} + 143 \, A a x^{4} + 117 \, {\left (B a + A b\right )} x^{5}\right )} e^{6} + 2 \, {\left (180 \, B b d x^{5} + 286 \, A a d x^{3} + 221 \, {\left (B a + A b\right )} d x^{4}\right )} e^{5} + 2 \, {\left (229 \, B b d^{2} x^{4} + 429 \, A a d^{2} x^{2} + 299 \, {\left (B a + A b\right )} d^{2} x^{3}\right )} e^{4} + 4 \, {\left (53 \, B b d^{3} x^{3} + 143 \, A a d^{3} x + 78 \, {\left (B a + A b\right )} d^{3} x^{2}\right )} e^{3} + {\left (3 \, B b d^{4} x^{2} + 143 \, A a d^{4} + 13 \, {\left (B a + A b\right )} d^{4} x\right )} e^{2} - 2 \, {\left (2 \, B b d^{5} x + 13 \, {\left (B a + A b\right )} 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.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 578 vs.
\(2 (87) = 174\).
time = 0.62, size = 578, normalized size = 6.96 \begin {gather*} \begin {cases} \frac {2 A a d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {4 A b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 A b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 A b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 A b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 A b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 A b e^{3} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 B b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 B b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 B b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 B b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 B b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 B b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 B b e^{3} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b 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. 1108 vs.
\(2 (75) = 150\).
time = 0.62, size = 1108, normalized size = 13.35 \begin {gather*} \frac {2}{45045} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d^{4} e^{\left (-1\right )} + 15015 \, {\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 b d^{4} e^{\left (-2\right )} + 12012 \, {\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 d^{3} e^{\left (-1\right )} + 12012 \, {\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 b d^{3} e^{\left (-2\right )} + 45045 \, \sqrt {x e + d} A a d^{4} + 60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a d^{3} + 7722 \, {\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 d^{2} e^{\left (-1\right )} + 7722 \, {\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 b 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 a d^{2} + 572 \, {\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 d e^{\left (-1\right )} + 572 \, {\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 b 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 a d + 65 \, {\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 a e^{\left (-1\right )} + 65 \, {\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 b 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 a\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,B\,b\,{\left (d+e\,x\right )}^2+143\,A\,a\,e^2+143\,B\,b\,d^2+117\,A\,b\,e\,\left (d+e\,x\right )+117\,B\,a\,e\,\left (d+e\,x\right )-234\,B\,b\,d\,\left (d+e\,x\right )-143\,A\,b\,d\,e-143\,B\,a\,d\,e\right )}{1287\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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