Optimal. Leaf size=218 \[ -\frac {2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac {2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \]
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Rubi [A] time = 0.10, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 77} \[ -\frac {2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac {2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e) \sqrt {d+e x}}{e^5}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^{3/2}}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^{5/2}}{e^5}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^{7/2}}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{9/2}}{e^5}+\frac {b^4 B (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4 (B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{5/2}}{5 e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{7/2}}{7 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{9/2}}{9 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{11/2}}{11 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 183, normalized size = 0.84 \[ \frac {2 (d+e x)^{3/2} \left (-4095 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+10010 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-12870 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+9009 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-15015 (b d-a e)^4 (B d-A e)+3465 b^4 B (d+e x)^5\right )}{45045 e^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 527, normalized size = 2.42 \[ \frac {2 \, {\left (3465 \, B b^{4} e^{6} x^{6} - 1280 \, B b^{4} d^{6} + 15015 \, A a^{4} d e^{5} + 1664 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e - 4576 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{2} + 6864 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{3} - 6006 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{4} + 315 \, {\left (B b^{4} d e^{5} + 13 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{4} d^{2} e^{4} - 13 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{5} - 286 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{6}\right )} x^{4} + 10 \, {\left (40 \, B b^{4} d^{3} e^{3} - 52 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{4} + 143 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{5} + 1287 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{6}\right )} x^{3} - 3 \, {\left (160 \, B b^{4} d^{4} e^{2} - 208 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{3} + 572 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{4} - 858 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{5} - 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{6}\right )} x^{2} + {\left (640 \, B b^{4} d^{5} e + 15015 \, A a^{4} e^{6} - 832 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{2} + 2288 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{3} - 3432 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{4} + 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 1144, normalized size = 5.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 469, normalized size = 2.15 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 b^{4} B \,x^{5} e^{5}+4095 A \,b^{4} e^{5} x^{4}+16380 B a \,b^{3} e^{5} x^{4}-3150 B \,b^{4} d \,e^{4} x^{4}+20020 A a \,b^{3} e^{5} x^{3}-3640 A \,b^{4} d \,e^{4} x^{3}+30030 B \,a^{2} b^{2} e^{5} x^{3}-14560 B a \,b^{3} d \,e^{4} x^{3}+2800 B \,b^{4} d^{2} e^{3} x^{3}+38610 A \,a^{2} b^{2} e^{5} x^{2}-17160 A a \,b^{3} d \,e^{4} x^{2}+3120 A \,b^{4} d^{2} e^{3} x^{2}+25740 B \,a^{3} b \,e^{5} x^{2}-25740 B \,a^{2} b^{2} d \,e^{4} x^{2}+12480 B a \,b^{3} d^{2} e^{3} x^{2}-2400 B \,b^{4} d^{3} e^{2} x^{2}+36036 A \,a^{3} b \,e^{5} x -30888 A \,a^{2} b^{2} d \,e^{4} x +13728 A a \,b^{3} d^{2} e^{3} x -2496 A \,b^{4} d^{3} e^{2} x +9009 B \,a^{4} e^{5} x -20592 B \,a^{3} b d \,e^{4} x +20592 B \,a^{2} b^{2} d^{2} e^{3} x -9984 B a \,b^{3} d^{3} e^{2} x +1920 B \,b^{4} d^{4} e x +15015 A \,a^{4} e^{5}-24024 A \,a^{3} b d \,e^{4}+20592 A \,a^{2} b^{2} d^{2} e^{3}-9152 A a \,b^{3} d^{3} e^{2}+1664 A \,b^{4} d^{4} e -6006 B \,a^{4} d \,e^{4}+13728 B \,d^{2} a^{3} b \,e^{3}-13728 B \,d^{3} a^{2} b^{2} e^{2}+6656 B a \,b^{3} d^{4} e -1280 B \,b^{4} d^{5}\right )}{45045 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 409, normalized size = 1.88 \[ \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{4} - 4095 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 197, normalized size = 0.90 \[ \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{11\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{5\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{7\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{9\,e^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.12, size = 517, normalized size = 2.37 \[ \frac {2 \left (\frac {B b^{4} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 6 A a^{2} b^{2} d^{2} e^{3} - 4 A a b^{3} d^{3} e^{2} + A b^{4} d^{4} e - B a^{4} d e^{4} + 4 B a^{3} b d^{2} e^{3} - 6 B a^{2} b^{2} d^{3} e^{2} + 4 B a b^{3} d^{4} e - B b^{4} d^{5}\right )}{3 e^{5}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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