Optimal. Leaf size=128 \[ -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]
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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)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 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)^{5/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{5/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{7/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^3}+\frac {b^2 B (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 138, normalized size = 1.08 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (143 a^2 e^2 (-2 B d+9 A e+7 B e x)+26 a b e \left (11 A e (-2 d+7 e x)+B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+b^2 \left (13 A e \left (8 d^2-28 d e x+63 e^2 x^2\right )+B \left (-48 d^3+168 d^2 e x-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 122, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(122\) |
default | \(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(122\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (693 b^{2} B \,x^{3} e^{3}+819 A \,b^{2} e^{3} x^{2}+1638 B a b \,e^{3} x^{2}-378 B \,b^{2} d \,e^{2} x^{2}+2002 A a b \,e^{3} x -364 A \,b^{2} d \,e^{2} x +1001 B \,a^{2} e^{3} x -728 B a b d \,e^{2} x +168 B \,b^{2} d^{2} e x +1287 a^{2} A \,e^{3}-572 A a b d \,e^{2}+104 A \,b^{2} d^{2} e -286 B \,a^{2} d \,e^{2}+208 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{9009 e^{4}}\) | \(169\) |
trager | \(\frac {2 \left (693 B \,b^{2} e^{6} x^{6}+819 A \,b^{2} e^{6} x^{5}+1638 B a b \,e^{6} x^{5}+1701 B \,b^{2} d \,e^{5} x^{5}+2002 A a b \,e^{6} x^{4}+2093 A \,b^{2} d \,e^{5} x^{4}+1001 B \,a^{2} e^{6} x^{4}+4186 B a b d \,e^{5} x^{4}+1113 B \,b^{2} d^{2} e^{4} x^{4}+1287 A \,a^{2} e^{6} x^{3}+5434 A a b d \,e^{5} x^{3}+1469 A \,b^{2} d^{2} e^{4} x^{3}+2717 B \,a^{2} d \,e^{5} x^{3}+2938 B a b \,d^{2} e^{4} x^{3}+15 B \,b^{2} d^{3} e^{3} x^{3}+3861 A \,a^{2} d \,e^{5} x^{2}+4290 A a b \,d^{2} e^{4} x^{2}+39 A \,b^{2} d^{3} e^{3} x^{2}+2145 B \,a^{2} d^{2} e^{4} x^{2}+78 B a b \,d^{3} e^{3} x^{2}-18 B \,b^{2} d^{4} e^{2} x^{2}+3861 A \,a^{2} d^{2} e^{4} x +286 A a b \,d^{3} e^{3} x -52 A \,b^{2} d^{4} e^{2} x +143 B \,a^{2} d^{3} e^{3} x -104 B a b \,d^{4} e^{2} x +24 B \,b^{2} d^{5} e x +1287 A \,a^{2} d^{3} e^{3}-572 A a b \,d^{4} e^{2}+104 A \,b^{2} d^{5} e -286 B \,a^{2} d^{4} e^{2}+208 B a b \,d^{5} e -48 B \,b^{2} d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(429\) |
risch | \(\frac {2 \left (693 B \,b^{2} e^{6} x^{6}+819 A \,b^{2} e^{6} x^{5}+1638 B a b \,e^{6} x^{5}+1701 B \,b^{2} d \,e^{5} x^{5}+2002 A a b \,e^{6} x^{4}+2093 A \,b^{2} d \,e^{5} x^{4}+1001 B \,a^{2} e^{6} x^{4}+4186 B a b d \,e^{5} x^{4}+1113 B \,b^{2} d^{2} e^{4} x^{4}+1287 A \,a^{2} e^{6} x^{3}+5434 A a b d \,e^{5} x^{3}+1469 A \,b^{2} d^{2} e^{4} x^{3}+2717 B \,a^{2} d \,e^{5} x^{3}+2938 B a b \,d^{2} e^{4} x^{3}+15 B \,b^{2} d^{3} e^{3} x^{3}+3861 A \,a^{2} d \,e^{5} x^{2}+4290 A a b \,d^{2} e^{4} x^{2}+39 A \,b^{2} d^{3} e^{3} x^{2}+2145 B \,a^{2} d^{2} e^{4} x^{2}+78 B a b \,d^{3} e^{3} x^{2}-18 B \,b^{2} d^{4} e^{2} x^{2}+3861 A \,a^{2} d^{2} e^{4} x +286 A a b \,d^{3} e^{3} x -52 A \,b^{2} d^{4} e^{2} x +143 B \,a^{2} d^{3} e^{3} x -104 B a b \,d^{4} e^{2} x +24 B \,b^{2} d^{5} e x +1287 A \,a^{2} d^{3} e^{3}-572 A a b \,d^{4} e^{2}+104 A \,b^{2} d^{5} e -286 B \,a^{2} d^{4} e^{2}+208 B a b \,d^{5} e -48 B \,b^{2} d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(429\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 168, normalized size = 1.31 \begin {gather*} \frac {2}{9009} \, {\left (693 \, {\left (x e + d\right )}^{\frac {13}{2}} B b^{2} - 819 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 1001 \, {\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 {9}{2}} - 1287 \, {\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 {7}{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. 344 vs.
\(2 (119) = 238\).
time = 0.87, size = 344, normalized size = 2.69 \begin {gather*} -\frac {2}{9009} \, {\left (48 \, B b^{2} d^{6} - {\left (693 \, B b^{2} x^{6} + 1287 \, A a^{2} x^{3} + 819 \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + 1001 \, {\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} e^{6} - {\left (1701 \, B b^{2} d x^{5} + 3861 \, A a^{2} d x^{2} + 2093 \, {\left (2 \, B a b + A b^{2}\right )} d x^{4} + 2717 \, {\left (B a^{2} + 2 \, A a b\right )} d x^{3}\right )} e^{5} - {\left (1113 \, B b^{2} d^{2} x^{4} + 3861 \, A a^{2} d^{2} x + 1469 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x^{3} + 2145 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} x^{2}\right )} e^{4} - {\left (15 \, B b^{2} d^{3} x^{3} + 1287 \, A a^{2} d^{3} + 39 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} x^{2} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} x\right )} e^{3} + 2 \, {\left (9 \, B b^{2} d^{4} x^{2} + 26 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} x + 143 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} e^{2} - 8 \, {\left (3 \, B b^{2} d^{5} x + 13 \, {\left (2 \, B a b + A b^{2}\right )} d^{5}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs.
\(2 (129) = 258\).
time = 0.49, size = 857, normalized size = 6.70 \begin {gather*} \begin {cases} \frac {2 A a^{2} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a^{2} d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a^{2} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a^{2} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 A a b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 A a b d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 A a b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 A a b d e x^{3} \sqrt {d + e x}}{63} + \frac {4 A a b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A b^{2} d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A b^{2} d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A b^{2} d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A b^{2} d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A b^{2} d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a^{2} d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a^{2} d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a^{2} d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a^{2} d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a^{2} e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {32 B a b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {16 B a b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {4 B a b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {452 B a b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {92 B a b d e x^{4} \sqrt {d + e x}}{99} + \frac {4 B a b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B b^{2} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B b^{2} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B b^{2} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B b^{2} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B b^{2} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B b^{2} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B b^{2} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\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. 1368 vs.
\(2 (119) = 238\).
time = 0.55, size = 1368, normalized size = 10.69 \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 = 1.21, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{11\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{9\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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