Optimal. Leaf size=146 \[ -\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45}
\begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^3 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^3 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^2}{(d+e x)^5} \, 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)^2}{e^2 (d+e x)^5}-\frac {2 b (b d-a e)}{e^2 (d+e x)^4}+\frac {b^2}{e^2 (d+e x)^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {2 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 73, normalized size = 0.50 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.05, size = 68, normalized size = 0.47
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (6 b^{2} e^{2} x^{2}+8 a b \,e^{2} x +4 b^{2} d e x +3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right )}{12 e^{3} \left (e x +d \right )^{4}}\) | \(68\) |
gosper | \(-\frac {\left (6 b^{2} e^{2} x^{2}+8 a b \,e^{2} x +4 b^{2} d e x +3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{12 e^{3} \left (e x +d \right )^{4} \left (b x +a \right )}\) | \(78\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{2} x^{2}}{2 e}-\frac {b \left (2 a e +b d \right ) x}{3 e^{2}}-\frac {3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}}{12 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{4}}\) | \(79\) |
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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Fricas [A]
time = 3.10, size = 89, normalized size = 0.61 \begin {gather*} -\frac {b^{2} d^{2} + {\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} e^{2} + 2 \, {\left (2 \, b^{2} d x + a b d\right )} e}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 104, normalized size = 0.71 \begin {gather*} \frac {- 3 a^{2} e^{2} - 2 a b d e - b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (- 8 a b e^{2} - 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 96, normalized size = 0.66 \begin {gather*} -\frac {{\left (6 \, b^{2} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, b^{2} d x e \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a b x e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.14, size = 77, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+8\,a\,b\,e^2\,x+b^2\,d^2+4\,b^2\,d\,e\,x+6\,b^2\,e^2\,x^2\right )}{12\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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