Optimal. Leaf size=39 \[ \frac {(a+b x) (d+e x)^5}{5 e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 32} \begin {gather*} \frac {(a+b x) (d+e x)^5}{5 e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^4}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(a+b x) (d+e x)^4}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int (d+e x)^4 \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) (d+e x)^5}{5 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.77 \begin {gather*} \frac {(a+b x) (d+e x)^5}{5 e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^4}{\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 = 0.42, size = 42, normalized size = 1.08 \begin {gather*} \frac {1}{5} \, e^{4} x^{5} + d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{3} + 2 \, d^{3} e x^{2} + d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 18, normalized size = 0.46 \begin {gather*} \frac {1}{5} \, {\left (x e + d\right )}^{5} e^{\left (-1\right )} \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 = 58, normalized size = 1.49 \begin {gather*} \frac {\left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \left (b x +a \right ) x}{5 \sqrt {\left (b x +a \right )^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 688, normalized size = 17.64 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{4} x^{4}}{5 \, b} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{4} x^{3}}{20 \, b^{2}} - \frac {77 \, a^{3} e^{4} x^{2}}{60 \, b^{3}} + \frac {47 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{4} x^{2}}{60 \, b^{3}} + \frac {77 \, a^{4} e^{4} x}{30 \, b^{4}} + \frac {a d^{4} \log \left (x + \frac {a}{b}\right )}{b} - \frac {a^{5} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {47 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4} e^{4}}{30 \, b^{5}} + \frac {{\left (4 \, b d e^{3} + a e^{4}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{3}}{4 \, b^{2}} + \frac {13 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{2} x^{2}}{12 \, b^{3}} - \frac {5 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a x^{2}}{3 \, b^{2}} + \frac {{\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} x^{2}}{b} - \frac {7 \, {\left (4 \, b d e^{3} + a e^{4}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x^{2}}{12 \, b^{3}} + \frac {2 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} - \frac {13 \, {\left (4 \, b d e^{3} + a e^{4}\right )} a^{3} x}{6 \, b^{4}} + \frac {10 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{2} x}{3 \, b^{3}} - \frac {2 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} a x}{b^{2}} + \frac {{\left (4 \, b d e^{3} + a e^{4}\right )} a^{4} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {2 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {2 \, {\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (b d^{4} + 4 \, a d^{3} e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {7 \, {\left (4 \, b d e^{3} + a e^{4}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{6 \, b^{5}} - \frac {4 \, {\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac {{\left (b d^{4} + 4 \, a d^{3} e\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}}{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 = -0.03 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 42, normalized size = 1.08 \begin {gather*} d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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