Optimal. Leaf size=208 \[ -\frac {2 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac {6 b (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}-\frac {2 b^2 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^4 (a+b x)} \]
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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 {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^4 (a+b x)}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^4 (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)^3}{5 e^4 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 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)^{3/2} \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)^{3/2} \, 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)^{3/2}}{e^3}+\frac {3 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{7/2}}{e^3}+\frac {b^6 (d+e x)^{9/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^3 (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 (b d-a e)^2 (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^2 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 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)^{5/2} \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 132, normalized size = 0.63
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 b^{3} x^{3} e^{3}+385 a \,b^{2} e^{3} x^{2}-70 b^{3} d \,e^{2} x^{2}+495 a^{2} b \,e^{3} x -220 a \,b^{2} d \,e^{2} x +40 b^{3} d^{2} e x +231 e^{3} a^{3}-198 a^{2} b d \,e^{2}+88 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1155 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 b^{3} x^{3} e^{3}+385 a \,b^{2} e^{3} x^{2}-70 b^{3} d \,e^{2} x^{2}+495 a^{2} b \,e^{3} x -220 a \,b^{2} d \,e^{2} x +40 b^{3} d^{2} e x +231 e^{3} a^{3}-198 a^{2} b d \,e^{2}+88 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{1155 e^{4} \left (b x +a \right )^{3}}\) | \(132\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (105 b^{3} x^{5} e^{5}+385 a \,b^{2} e^{5} x^{4}+140 b^{3} d \,e^{4} x^{4}+495 a^{2} b \,e^{5} x^{3}+550 a \,b^{2} d \,e^{4} x^{3}+5 b^{3} d^{2} e^{3} x^{3}+231 a^{3} e^{5} x^{2}+792 a^{2} b d \,e^{4} x^{2}+33 a \,b^{2} d^{2} e^{3} x^{2}-6 b^{3} d^{3} e^{2} x^{2}+462 a^{3} d \,e^{4} x +99 a^{2} b \,d^{2} e^{3} x -44 a \,b^{2} d^{3} e^{2} x +8 b^{3} d^{4} e x +231 a^{3} d^{2} e^{3}-198 a^{2} b \,d^{3} e^{2}+88 a \,b^{2} d^{4} e -16 b^{3} d^{5}\right ) \sqrt {e x +d}}{1155 \left (b x +a \right ) e^{4}}\) | \(244\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 203, normalized size = 0.98 \begin {gather*} \frac {2}{1155} \, {\left (105 \, b^{3} x^{5} e^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d 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 = 1.92, size = 204, normalized size = 0.98 \begin {gather*} -\frac {2}{1155} \, {\left (16 \, b^{3} d^{5} - {\left (105 \, b^{3} x^{5} + 385 \, a b^{2} x^{4} + 495 \, a^{2} b x^{3} + 231 \, a^{3} x^{2}\right )} e^{5} - 2 \, {\left (70 \, b^{3} d x^{4} + 275 \, a b^{2} d x^{3} + 396 \, a^{2} b d x^{2} + 231 \, a^{3} d x\right )} e^{4} - {\left (5 \, b^{3} d^{2} x^{3} + 33 \, a b^{2} d^{2} x^{2} + 99 \, a^{2} b d^{2} x + 231 \, a^{3} d^{2}\right )} e^{3} + 2 \, {\left (3 \, b^{3} d^{3} x^{2} + 22 \, a b^{2} d^{3} x + 99 \, a^{2} b d^{3}\right )} e^{2} - 8 \, {\left (b^{3} d^{4} x + 11 \, a b^{2} d^{4}\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 {3}{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. 670 vs.
\(2 (151) = 302\).
time = 1.08, size = 670, normalized size = 3.22 \begin {gather*} \frac {2}{3465} \, {\left (3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b d^{2} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 693 \, {\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^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 99 \, {\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^{2} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 1386 \, {\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 e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 594 \, {\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 e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 22 \, {\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 e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 3465 \, \sqrt {x e + d} a^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} d \mathrm {sgn}\left (b x + a\right ) + 297 \, {\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 e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 33 \, {\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} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\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} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 231 \, {\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} \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 )}^{3/2}\,{\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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