Optimal. Leaf size=208 \[ -\frac {2 (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac {2 b (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}-\frac {6 b^2 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^4 (a+b x)} \]
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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)^{11/2} (b d-a e)}{11 e^4 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^4 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^4 (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)^{5/2} \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 (d+e x)^{5/2} \, 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 (d+e x)^{5/2}}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{7/2}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{9/2}}{e^3}+\frac {b^6 (d+e x)^{11/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^3 (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 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}-\frac {6 b^2 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^4 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 120, normalized size = 0.58 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (429 a^3 e^3+143 a^2 b e^2 (-2 d+7 e x)+13 a b^2 e \left (8 d^2-28 d e x+63 e^2 x^2\right )+b^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 132, normalized size = 0.63
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 b^{3} x^{3} e^{3}+819 a \,b^{2} e^{3} x^{2}-126 b^{3} d \,e^{2} x^{2}+1001 a^{2} b \,e^{3} x -364 a \,b^{2} d \,e^{2} x +56 b^{3} d^{2} e x +429 e^{3} a^{3}-286 a^{2} b d \,e^{2}+104 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3003 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 b^{3} x^{3} e^{3}+819 a \,b^{2} e^{3} x^{2}-126 b^{3} d \,e^{2} x^{2}+1001 a^{2} b \,e^{3} x -364 a \,b^{2} d \,e^{2} x +56 b^{3} d^{2} e x +429 e^{3} a^{3}-286 a^{2} b d \,e^{2}+104 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3003 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (231 e^{6} b^{3} x^{6}+819 a \,b^{2} e^{6} x^{5}+567 b^{3} d \,e^{5} x^{5}+1001 a^{2} b \,e^{6} x^{4}+2093 a \,b^{2} d \,e^{5} x^{4}+371 b^{3} d^{2} e^{4} x^{4}+429 a^{3} e^{6} x^{3}+2717 a^{2} b d \,e^{5} x^{3}+1469 a \,b^{2} d^{2} e^{4} x^{3}+5 b^{3} d^{3} e^{3} x^{3}+1287 a^{3} d \,e^{5} x^{2}+2145 a^{2} b \,d^{2} e^{4} x^{2}+39 a \,b^{2} d^{3} e^{3} x^{2}-6 b^{3} d^{4} e^{2} x^{2}+1287 a^{3} d^{2} e^{4} x +143 a^{2} b \,d^{3} e^{3} x -52 a \,b^{2} d^{4} e^{2} x +8 b^{3} d^{5} e x +429 a^{3} d^{3} e^{3}-286 a^{2} b \,d^{4} e^{2}+104 a \,b^{2} d^{5} e -16 b^{3} d^{6}\right ) \sqrt {e x +d}}{3003 \left (b x +a \right ) e^{4}}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 251, normalized size = 1.21 \begin {gather*} \frac {2}{3003} \, {\left (231 \, b^{3} x^{6} e^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} 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 = 2.72, size = 255, normalized size = 1.23 \begin {gather*} -\frac {2}{3003} \, {\left (16 \, b^{3} d^{6} - {\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} e^{6} - {\left (567 \, b^{3} d x^{5} + 2093 \, a b^{2} d x^{4} + 2717 \, a^{2} b d x^{3} + 1287 \, a^{3} d x^{2}\right )} e^{5} - {\left (371 \, b^{3} d^{2} x^{4} + 1469 \, a b^{2} d^{2} x^{3} + 2145 \, a^{2} b d^{2} x^{2} + 1287 \, a^{3} d^{2} x\right )} e^{4} - {\left (5 \, b^{3} d^{3} x^{3} + 39 \, a b^{2} d^{3} x^{2} + 143 \, a^{2} b d^{3} x + 429 \, a^{3} d^{3}\right )} e^{3} + 2 \, {\left (3 \, b^{3} d^{4} x^{2} + 26 \, a b^{2} d^{4} x + 143 \, a^{2} b d^{4}\right )} e^{2} - 8 \, {\left (b^{3} d^{5} x + 13 \, a b^{2} d^{5}\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 \left (d + e x\right )^{\frac {5}{2}} \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. 1004 vs.
\(2 (151) = 302\).
time = 1.52, size = 1004, normalized size = 4.83 \begin {gather*} \frac {2}{15015} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d^{3} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 3003 \, {\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^{3} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 429 \, {\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^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\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 d^{2} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 3861 \, {\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} d^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 143 \, {\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} d^{2} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 15015 \, \sqrt {x e + d} a^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) + 3861 \, {\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^{2} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 429 \, {\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 )} a b^{2} d e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{3} d e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 3003 \, {\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^{3} d \mathrm {sgn}\left (b x + a\right ) + 143 \, {\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 )} a^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 429 \, {\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^{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 {\left (d+e\,x\right )}^{5/2}\,{\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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