Optimal. Leaf size=114 \[ \frac {(b d-a e)^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3}+\frac {2 e (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac {e^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.10, number of steps
used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {659}
\begin {gather*} \frac {2 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)}{5 b^3}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^2}{4 b^3}+\frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 659
Rubi steps
\begin {align*} \int (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 \left (\frac {(b d-a e)^2 \left (a b+b^2 x\right )^3}{b^2}+\frac {2 e (b d-a e) \left (a b+b^2 x\right )^4}{b^3}+\frac {e^2 \left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^3}+\frac {2 e (b d-a e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac {e^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 127, normalized size = 1.11 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (20 a^3 \left (3 d^2+3 d e x+e^2 x^2\right )+15 a^2 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+6 a b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )}{60 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 148, normalized size = 1.30
method | result | size |
gosper | \(\frac {x \left (10 b^{3} e^{2} x^{5}+36 x^{4} a \,b^{2} e^{2}+24 x^{4} b^{3} d e +45 x^{3} e^{2} a^{2} b +90 x^{3} d e a \,b^{2}+15 x^{3} b^{3} d^{2}+20 x^{2} a^{3} e^{2}+120 x^{2} a^{2} b d e +60 x^{2} a \,b^{2} d^{2}+60 x d e \,a^{3}+90 a^{2} b \,d^{2} x +60 d^{2} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(148\) |
default | \(\frac {x \left (10 b^{3} e^{2} x^{5}+36 x^{4} a \,b^{2} e^{2}+24 x^{4} b^{3} d e +45 x^{3} e^{2} a^{2} b +90 x^{3} d e a \,b^{2}+15 x^{3} b^{3} d^{2}+20 x^{2} a^{3} e^{2}+120 x^{2} a^{2} b d e +60 x^{2} a \,b^{2} d^{2}+60 x d e \,a^{3}+90 a^{2} b \,d^{2} x +60 d^{2} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3}}\) | \(148\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{3} e^{2} x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 a \,b^{2} e^{2}+2 b^{3} d e \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 e^{2} a^{2} b +6 d e a \,b^{2}+b^{3} d^{2}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{3} e^{2}+6 a^{2} b d e +3 a \,b^{2} d^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (2 d e \,a^{3}+3 a^{2} b \,d^{2}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{3} d^{2} x}{b x +a}\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 244 vs.
\(2 (104) = 208\).
time = 0.34, size = 244, normalized size = 2.14 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} d^{2} x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d x e}{2 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d^{2}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x e^{2}}{4 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d e}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e^{2}}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} x e^{2}}{6 \, b^{2}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d e}{5 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e^{2}}{30 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.93, size = 125, normalized size = 1.10 \begin {gather*} \frac {1}{4} \, b^{3} d^{2} x^{4} + a b^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x + \frac {1}{60} \, {\left (10 \, b^{3} x^{6} + 36 \, a b^{2} x^{5} + 45 \, a^{2} b x^{4} + 20 \, a^{3} x^{3}\right )} e^{2} + \frac {1}{10} \, {\left (4 \, b^{3} d x^{5} + 15 \, a b^{2} d x^{4} + 20 \, a^{2} b d x^{3} + 10 \, a^{3} d x^{2}\right )} e \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 {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.81, size = 202, normalized size = 1.77 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, b^{3} d x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, b^{3} d^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a b^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a b^{2} d x^{4} e \mathrm {sgn}\left (b x + a\right ) + a b^{2} d^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, a^{2} b x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b d x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b d^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + a^{3} d x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{3} 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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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 )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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