Optimal. Leaf size=125 \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{3 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3} \]
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Rubi [A] time = 0.13, antiderivative size = 125, 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} \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{3 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^3} \]
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)^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^3 (d+e x)^2 \, dx}{b^2 \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)^4 (d+e x)^2 \, 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)^2 (a+b x)^4}{b^2}+\frac {2 e (b d-a e) (a+b x)^5}{b^2}+\frac {e^2 (a+b x)^6}{b^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac {e (b d-a e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e^2 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 157, normalized size = 1.26 \[ \frac {x \sqrt {(a+b x)^2} \left (35 a^4 \left (3 d^2+3 d e x+e^2 x^2\right )+35 a^3 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+21 a^2 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+7 a b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )}{105 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 156, normalized size = 1.25 \[ \frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 260, normalized size = 2.08 \[ \frac {1}{7} \, b^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{4} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{5} \, a b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, a^{2} b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{3} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a^{3} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{4} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{4} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d^{2} x \mathrm {sgn}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 189, normalized size = 1.51 \[ \frac {\left (15 e^{2} b^{4} x^{6}+70 x^{5} e^{2} a \,b^{3}+35 x^{5} b^{4} d e +126 x^{4} a^{2} b^{2} e^{2}+168 x^{4} d e a \,b^{3}+21 x^{4} b^{4} d^{2}+105 a^{3} b \,e^{2} x^{3}+315 a^{2} b^{2} d e \,x^{3}+105 a \,b^{3} d^{2} x^{3}+35 x^{2} e^{2} a^{4}+280 x^{2} d e \,a^{3} b +210 x^{2} d^{2} a^{2} b^{2}+105 a^{4} d e x +210 a^{3} b \,d^{2} x +105 d^{2} a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{105 \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 452, normalized size = 3.62 \[ \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{2} x}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e^{2} x^{2}}{7 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d^{2}}{4 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4} e^{2}}{4 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{2} x}{14 \, b^{2}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{2}}{70 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{4 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{4 \, b^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} x}{6 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (2 \, b d e + a e^{2}\right )} a}{30 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d^{2} + 2 \, a d e\right )}}{5 \, b^{2}} \]
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 \[ \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (a + b x\right ) \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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