Optimal. Leaf size=208 \[ -\frac {2 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac {6 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}-\frac {6 b^2 (b d-a e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 208, 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 {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 \sqrt {d+e x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 \sqrt {d+e x}}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{5/2}}{e^3}+\frac {b^6 (d+e x)^{7/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac {6 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}-\frac {6 b^2 (b d-a e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 120, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (105 a^3 e^3+63 a^2 b e^2 (-2 d+3 e x)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 132, normalized size = 0.63
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{3} x^{3} e^{3}+135 a \,b^{2} e^{3} x^{2}-30 b^{3} d \,e^{2} x^{2}+189 a^{2} b \,e^{3} x -108 a \,b^{2} d \,e^{2} x +24 b^{3} d^{2} e x +105 e^{3} a^{3}-126 a^{2} b d \,e^{2}+72 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{3} x^{3} e^{3}+135 a \,b^{2} e^{3} x^{2}-30 b^{3} d \,e^{2} x^{2}+189 a^{2} b \,e^{3} x -108 a \,b^{2} d \,e^{2} x +24 b^{3} d^{2} e x +105 e^{3} a^{3}-126 a^{2} b d \,e^{2}+72 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (35 b^{3} x^{4} e^{4}+135 a \,b^{2} e^{4} x^{3}+5 b^{3} d \,e^{3} x^{3}+189 a^{2} b \,e^{4} x^{2}+27 a \,b^{2} d \,e^{3} x^{2}-6 b^{3} d^{2} e^{2} x^{2}+105 a^{3} e^{4} x +63 a^{2} b d \,e^{3} x -36 a \,b^{2} d^{2} e^{2} x +8 b^{3} d^{3} e x +105 a^{3} d \,e^{3}-126 a^{2} b \,d^{2} e^{2}+72 a \,b^{2} d^{3} e -16 b^{3} d^{4}\right ) \sqrt {e x +d}}{315 \left (b x +a \right ) e^{4}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 155, normalized size = 0.75 \begin {gather*} \frac {2}{315} \, {\left (35 \, b^{3} x^{4} e^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \, {\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} + {\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.24, size = 152, normalized size = 0.73 \begin {gather*} -\frac {2}{315} \, {\left (16 \, b^{3} d^{4} - {\left (35 \, b^{3} x^{4} + 135 \, a b^{2} x^{3} + 189 \, a^{2} b x^{2} + 105 \, a^{3} x\right )} e^{4} - {\left (5 \, b^{3} d x^{3} + 27 \, a b^{2} d x^{2} + 63 \, a^{2} b d x + 105 \, a^{3} d\right )} e^{3} + 6 \, {\left (b^{3} d^{2} x^{2} + 6 \, a b^{2} d^{2} x + 21 \, a^{2} b d^{2}\right )} e^{2} - 8 \, {\left (b^{3} d^{3} x + 9 \, a b^{2} d^{3}\right )} e\right )} \sqrt {x e + d} e^{\left (-4\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 \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{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. 387 vs.
\(2 (151) = 302\).
time = 1.01, size = 387, normalized size = 1.86 \begin {gather*} \frac {2}{315} \, {\left (315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\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 b^{2} d e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 9 \, {\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^{3} d e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 63 \, {\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^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 27 \, {\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 )} a b^{2} e^{\left (-2\right )} \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 )} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} a^{3} d \mathrm {sgn}\left (b x + a\right ) + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} \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 = -0.00 \begin {gather*} \int \sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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