Optimal. Leaf size=41 \[ \frac {2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 32}
\begin {gather*} \frac {2 (a+b x) (d+e x)^{9/2}}{9 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 784
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^{7/2}}{\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)^{7/2}}{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)^{7/2} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 0.78 \begin {gather*} \frac {2 (a+b x) (d+e x)^{9/2}}{9 e \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 27, normalized size = 0.66
method | result | size |
gosper | \(\frac {2 \left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e \sqrt {\left (b x +a \right )^{2}}}\) | \(27\) |
default | \(\frac {2 \left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e \sqrt {\left (b x +a \right )^{2}}}\) | \(27\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (e^{4} x^{4}+4 d \,x^{3} e^{3}+6 d^{2} x^{2} e^{2}+4 d^{3} x e +d^{4}\right ) \sqrt {e x +d}}{9 \left (b x +a \right ) e}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 48, normalized size = 1.17 \begin {gather*} \frac {2}{9} \, {\left (x^{4} e^{4} + 4 \, d x^{3} e^{3} + 6 \, d^{2} x^{2} e^{2} + 4 \, d^{3} x e + d^{4}\right )} \sqrt {x e + d} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.19, size = 48, normalized size = 1.17 \begin {gather*} \frac {2}{9} \, {\left (x^{4} e^{4} + 4 \, d x^{3} e^{3} + 6 \, d^{2} x^{2} e^{2} + 4 \, d^{3} x e + d^{4}\right )} \sqrt {x e + d} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs.
\(2 (26) = 52\).
time = 1.34, size = 226, normalized size = 5.51 \begin {gather*} \frac {2}{315} \, {\left (315 \, \sqrt {x e + d} d^{4} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} d^{3} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} d^{2} \mathrm {sgn}\left (b x + a\right ) + 36 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} d \mathrm {sgn}\left (b x + a\right ) + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \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 = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{\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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