Optimal. Leaf size=42 \[ \frac {-a-b x}{3 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 0.93, 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}{3 e \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3} \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}{\left (a b+b^2 x\right ) (d+e x)^4} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(d+e x)^4} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a+b x}{3 e (d+e x)^3 \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 = 0.71 \begin {gather*} -\frac {a+b x}{3 e \sqrt {(a+b x)^2} (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 1.03, 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 = 0.47, size = 35, normalized size = 0.83 \begin {gather*} -\frac {1}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 18, normalized size = 0.43 \begin {gather*} -\frac {e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.64 \begin {gather*} -\frac {b x +a}{3 \left (e x +d \right )^{3} \sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 28, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}}{3\,e\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 37, normalized size = 0.88 \begin {gather*} - \frac {1}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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