Optimal. Leaf size=125 \[ \frac {2 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{6 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \]
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Rubi [A] time = 0.14, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {2 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^2}{6 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^2 \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 \left (a b+b^2 x\right )^5 (d+e x)^2 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(b d-a e)^2 \left (a b+b^2 x\right )^5}{b^2}+\frac {2 e (b d-a e) \left (a b+b^2 x\right )^6}{b^3}+\frac {e^2 \left (a b+b^2 x\right )^7}{b^4}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}+\frac {2 e (b d-a e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^3}+\frac {e^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 187, normalized size = 1.50 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (56 a^5 \left (3 d^2+3 d e x+e^2 x^2\right )+70 a^4 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+56 a^3 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+28 a^2 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+8 a b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )}{168 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.74, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 197, normalized size = 1.58 \begin {gather*} \frac {1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac {1}{7} \, {\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} + {\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 320, normalized size = 2.56 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, b^{5} d x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a b^{4} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a b^{4} d x^{6} e \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{2} b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{2} b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{4} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{4} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{5} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{5} 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 [B] time = 0.04, size = 230, normalized size = 1.84 \begin {gather*} \frac {\left (21 e^{2} b^{5} x^{7}+120 x^{6} e^{2} a \,b^{4}+48 x^{6} d e \,b^{5}+280 x^{5} e^{2} a^{2} b^{3}+280 x^{5} d e a \,b^{4}+28 x^{5} d^{2} b^{5}+336 a^{3} b^{2} e^{2} x^{4}+672 a^{2} b^{3} d e \,x^{4}+168 a \,b^{4} d^{2} x^{4}+210 x^{3} e^{2} a^{4} b +840 x^{3} d e \,a^{3} b^{2}+420 x^{3} d^{2} a^{2} b^{3}+56 x^{2} e^{2} a^{5}+560 x^{2} d e \,a^{4} b +560 x^{2} d^{2} a^{3} b^{2}+168 x d e \,a^{5}+420 x \,d^{2} a^{4} b +168 d^{2} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{168 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 245, normalized size = 1.96 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d e x}{3 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e^{2} x}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{2}}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d e}{3 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e^{2}}{6 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{2} x}{8 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e}{7 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{2}}{56 \, b^{3}} \end {gather*}
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 \begin {gather*} \int {\left (d+e\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
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 \begin {gather*} \int \left (d + e x\right )^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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