Optimal. Leaf size=96 \[ -\frac {2 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {2 b (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^2 (a+b x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45}
\begin {gather*} \frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^2 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^2 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (d+e x)^{3/2} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (d+e x)^{3/2}}{e}+\frac {b^2 (d+e x)^{5/2}}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}+\frac {2 b (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^2 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.50 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} (-2 b d+7 a e+5 b e x)}{35 e^2 (a+b x)} \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 = 0.48, size = 33, normalized size = 0.34
method | result | size |
default | \(\frac {2 \,\mathrm {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {5}{2}} \left (5 b e x +7 a e -2 b d \right )}{35 e^{2}}\) | \(33\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 b e x +7 a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{35 e^{2} \left (b x +a \right )}\) | \(43\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (5 b \,e^{3} x^{3}+7 a \,e^{3} x^{2}+8 b d \,e^{2} x^{2}+14 a d \,e^{2} x +b \,d^{2} e x +7 a \,d^{2} e -2 b \,d^{3}\right ) \sqrt {e x +d}}{35 \left (b x +a \right ) e^{2}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 67, normalized size = 0.70 \begin {gather*} \frac {2}{35} \, {\left (5 \, b x^{3} e^{3} - 2 \, b d^{3} + 7 \, a d^{2} e + {\left (8 \, b d e^{2} + 7 \, a e^{3}\right )} x^{2} + {\left (b d^{2} e + 14 \, a d e^{2}\right )} x\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.13, size = 70, normalized size = 0.73 \begin {gather*} -\frac {2}{35} \, {\left (2 \, b d^{3} - {\left (5 \, b x^{3} + 7 \, a x^{2}\right )} e^{3} - 2 \, {\left (4 \, b d x^{2} + 7 \, a d x\right )} e^{2} - {\left (b d^{2} x + 7 \, a d^{2}\right )} e\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{\frac {3}{2}} \sqrt {\left (a + b x\right )^{2}}\, dx \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. 239 vs.
\(2 (67) = 134\).
time = 1.31, size = 239, normalized size = 2.49 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d^{2} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 14 \, {\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 )} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {x e + d} a d^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d \mathrm {sgn}\left (b x + a\right ) + 3 \, {\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 )} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 7 \, {\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 )} a \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 = -0.01 \begin {gather*} \int \sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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