Optimal. Leaf size=172 \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]
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Rubi [A] time = 0.24, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)}{3 b^4}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2}{8 b^4}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3}{7 b^4}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^4} \]
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^3 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^3 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(b d-a e)^3 (a+b x)^6}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^7}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^8}{b^3}+\frac {e^3 (a+b x)^9}{b^3}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {e^2 (b d-a e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^4}+\frac {e^3 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 294, normalized size = 1.71 \[ \frac {x \sqrt {(a+b x)^2} \left (210 a^6 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+252 a^5 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+210 a^4 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+10 a b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )}{840 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 327, normalized size = 1.90 \[ \frac {1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac {1}{3} \, {\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 523, normalized size = 3.04 \[ \frac {1}{10} \, b^{6} x^{10} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{6} d x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, b^{6} d^{2} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a b^{5} x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, a b^{5} d x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{7} \, a b^{5} d^{2} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{8} \, a^{2} b^{4} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {45}{7} \, a^{2} b^{4} d x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{4} d^{2} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, a^{3} b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{4} b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {45}{4} \, a^{4} b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, a^{5} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{5} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{6} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{6} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{3} x \mathrm {sgn}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 380, normalized size = 2.21 \[ \frac {\left (84 e^{3} b^{6} x^{9}+560 x^{8} e^{3} a \,b^{5}+280 x^{8} d \,e^{2} b^{6}+1575 x^{7} e^{3} a^{2} b^{4}+1890 x^{7} d \,e^{2} a \,b^{5}+315 x^{7} d^{2} e \,b^{6}+2400 x^{6} e^{3} a^{3} b^{3}+5400 x^{6} d \,e^{2} a^{2} b^{4}+2160 x^{6} d^{2} e a \,b^{5}+120 x^{6} d^{3} b^{6}+2100 x^{5} e^{3} a^{4} b^{2}+8400 x^{5} d \,e^{2} a^{3} b^{3}+6300 x^{5} d^{2} e \,a^{2} b^{4}+840 x^{5} d^{3} a \,b^{5}+1008 x^{4} e^{3} a^{5} b +7560 x^{4} d \,e^{2} a^{4} b^{2}+10080 x^{4} d^{2} e \,a^{3} b^{3}+2520 x^{4} d^{3} a^{2} b^{4}+210 x^{3} e^{3} a^{6}+3780 x^{3} d \,e^{2} a^{5} b +9450 x^{3} d^{2} e \,a^{4} b^{2}+4200 x^{3} d^{3} a^{3} b^{3}+840 a^{6} d \,e^{2} x^{2}+5040 a^{5} b \,d^{2} e \,x^{2}+4200 a^{4} b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{6}+2520 x \,d^{3} a^{5} b +840 d^{3} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{840 \left (b x +a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 693, normalized size = 4.03 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{3} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{3} x}{6 \, b^{3}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{3} x^{2}}{90 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{3}}{6 \, b^{4}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{3} x}{180 \, b^{3}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{3}}{1260 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{6 \, b^{3}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{6 \, b^{4}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} - \frac {11 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{72 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} + \frac {83 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{504 \, b^{4}} - \frac {27 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right ) \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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