Optimal. Leaf size=125 \[ \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \]
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Rubi [A] time = 0.10, 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} \begin {gather*} \frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \end {gather*}
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 \sqrt {a^2+2 a b x+b^2 x^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 ) (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 (a+b x)^2 (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)^2}{b^2}+\frac {2 e (b d-a e) (a+b x)^3}{b^2}+\frac {e^2 (a+b x)^4}{b^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^3}+\frac {e (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^3}+\frac {e^2 (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 97, normalized size = 0.78 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (10 a^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )}{30 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 81, normalized size = 0.65 \begin {gather*} \frac {1}{5} \, b^{2} e^{2} x^{5} + a^{2} d^{2} x + \frac {1}{2} \, {\left (b^{2} d e + a b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{3} + {\left (a b d^{2} + a^{2} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 143, normalized size = 1.14 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, a b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + a b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{2} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{2} d^{2} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.86 \begin {gather*} \frac {\left (6 b^{2} e^{2} x^{4}+15 x^{3} a b \,e^{2}+15 x^{3} b^{2} d e +10 x^{2} a^{2} e^{2}+40 x^{2} a b d e +10 x^{2} b^{2} d^{2}+30 a^{2} d e x +30 a b \,d^{2} x +30 a^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{30 b x +30 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 452, normalized size = 3.62 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d^{2} x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{2} x}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e^{2} x^{2}}{5 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d^{2}}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{2}}{2 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a e^{2} x}{20 \, b^{2}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e^{2}}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{2} x}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a x}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (2 \, b d e + a e^{2}\right )} a^{3}}{2 \, b^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d^{2} + 2 \, a d e\right )} a^{2}}{2 \, b^{2}} + \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 )} x}{4 \, b^{2}} - \frac {5 \, {\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}{12 \, 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 )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 438, normalized size = 3.50 \begin {gather*} a\,d^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}+\frac {d^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {e^2\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b}-\frac {11\,a^2\,e^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{160\,b^5}-\frac {a^3\,e^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2}+\frac {d\,e\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{2\,b}-\frac {7\,a\,e^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{60\,b^3}+\frac {a\,e^2\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {a^2\,d\,e\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}-\frac {a\,d\,e\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{48\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 87, normalized size = 0.70 \begin {gather*} a^{2} d^{2} x + \frac {b^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac {a b e^{2}}{2} + \frac {b^{2} d e}{2}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {4 a b d e}{3} + \frac {b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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