Optimal. Leaf size=78 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)}{7 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 b^2} \]
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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)^6 (b d-a e)}{7 b^2}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 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 )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x) \, dx}{b^4 \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)^6 (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)^6}{b}+\frac {e (a+b x)^7}{b}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^2}+\frac {e (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 140, normalized size = 1.79 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (28 a^6 (2 d+e x)+56 a^5 b x (3 d+2 e x)+70 a^4 b^2 x^2 (4 d+3 e x)+56 a^3 b^3 x^3 (5 d+4 e x)+28 a^2 b^4 x^4 (6 d+5 e x)+8 a b^5 x^5 (7 d+6 e x)+b^6 x^6 (8 d+7 e x)\right )}{56 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.32, 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 )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 142, normalized size = 1.82 \begin {gather*} \frac {1}{8} \, b^{6} e x^{8} + a^{6} d x + \frac {1}{7} \, {\left (b^{6} d + 6 \, a b^{5} e\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d + 5 \, a^{2} b^{4} e\right )} x^{6} + {\left (3 \, a^{2} b^{4} d + 4 \, a^{3} b^{3} e\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d + 3 \, a^{4} b^{2} e\right )} x^{4} + {\left (5 \, a^{4} b^{2} d + 2 \, a^{5} b e\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d + a^{6} 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 = 236, normalized size = 3.03 \begin {gather*} \frac {1}{8} \, b^{6} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, a b^{5} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} b^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{4} \, a^{4} b^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{5} b x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{6} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} 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 = 162, normalized size = 2.08 \begin {gather*} \frac {\left (7 e \,b^{6} x^{7}+48 x^{6} e a \,b^{5}+8 x^{6} d \,b^{6}+140 x^{5} e \,a^{2} b^{4}+56 x^{5} d a \,b^{5}+224 a^{3} b^{3} e \,x^{4}+168 a^{2} b^{4} d \,x^{4}+210 x^{3} e \,a^{4} b^{2}+280 x^{3} d \,a^{3} b^{3}+112 a^{5} b e \,x^{2}+280 a^{4} b^{2} d \,x^{2}+28 x e \,a^{6}+168 x d \,a^{5} b +56 d \,a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{56 \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 = 0.63, size = 251, normalized size = 3.22 \begin {gather*} \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} e}{6 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d + a e\right )} a x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e x}{8 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (b d + a e\right )} a^{2}}{6 \, b^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e}{56 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} {\left (b d + a e\right )}}{7 \, 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 )}^{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 (a + b x\right ) \left (d + e x\right ) \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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