Optimal. Leaf size=39 \[ \frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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)^4}{4 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)^3}{\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)^3}{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)^3 \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(a+b x) (d+e x)^4}{4 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} \frac {(a+b x) (d+e x)^4}{4 e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^3}{\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 = 31, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, e^{3} x^{4} + d e^{2} x^{3} + \frac {3}{2} \, d^{2} e x^{2} + d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 18, normalized size = 0.46 \begin {gather*} \frac {1}{4} \, {\left (x e + d\right )}^{4} 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 [A] time = 0.05, size = 47, normalized size = 1.21 \begin {gather*} \frac {\left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (b x +a \right ) x}{4 \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 = 0.53, size = 432, normalized size = 11.08 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{3}}{4 \, b} + \frac {13 \, a^{2} e^{3} x^{2}}{12 \, b^{2}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{3} x^{2}}{12 \, b^{2}} - \frac {13 \, a^{3} e^{3} x}{6 \, b^{3}} + \frac {a d^{3} \log \left (x + \frac {a}{b}\right )}{b} + \frac {a^{4} e^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{3}}{6 \, b^{4}} - \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a x^{2}}{6 \, b^{2}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} x^{2}}{2 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} + \frac {5 \, {\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{3 \, b^{3}} - \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a x}{b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {2 \, {\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac {{\left (b d^{3} + 3 \, a d^{2} 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 )}^3}{\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 = 0.11, size = 32, normalized size = 0.82 \begin {gather*} d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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