Optimal. Leaf size=78 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)}{5 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {770, 21, 43} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)}{5 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^2} \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) \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) \, 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) \, 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) (a+b x)^4}{b}+\frac {e (a+b x)^5}{b}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) (a+b x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{5 b^2}+\frac {e (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 102, normalized size = 1.31 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (15 a^4 (2 d+e x)+20 a^3 b x (3 d+2 e x)+15 a^2 b^2 x^2 (4 d+3 e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right )}{30 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.01, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 96, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac {1}{5} \, {\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 162, normalized size = 2.08 \begin {gather*} \frac {1}{6} \, b^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, a b^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + a b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, a^{3} b x^{3} e \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{4} d 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 = 0.05, size = 114, normalized size = 1.46 \begin {gather*} \frac {\left (5 e \,b^{4} x^{5}+24 x^{4} e a \,b^{3}+6 x^{4} d \,b^{4}+45 x^{3} e \,a^{2} b^{2}+30 x^{3} d a \,b^{3}+40 x^{2} e \,a^{3} b +60 x^{2} d \,a^{2} b^{2}+15 a^{4} e x +60 a^{3} b d x +30 d \,a^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{30 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 251, normalized size = 3.22 \begin {gather*} \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a d x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} e x}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} d}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} e}{4 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d + a e\right )} a x}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} e x}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (b d + a e\right )} a^{2}}{4 \, b^{2}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e}{30 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d + a e\right )}}{5 \, b^{2}} \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 (a+b\,x\right )\,\left (d+e\,x\right )\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right ) \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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