Optimal. Leaf size=204 \[ -\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^4 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^4 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^4 (a+b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 204, 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 = {646, 43} \[ \frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^4 (a+b x)}-\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^4 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^4 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\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}{e^3 \sqrt {d+e x}}+\frac {3 b^4 (b d-a e)^2 \sqrt {d+e x}}{e^3}-\frac {3 b^5 (b d-a e) (d+e x)^{3/2}}{e^3}+\frac {b^6 (d+e x)^{5/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}+\frac {2 b (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^4 (a+b x)}-\frac {6 b^2 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}+\frac {2 b^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 119, normalized size = 0.58 \[ \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (35 a^3 e^3+35 a^2 b e^2 (e x-2 d)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 115, normalized size = 0.56 \[ \frac {2 \, {\left (5 \, b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e - 70 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} - 3 \, {\left (2 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (8 \, b^{3} d^{2} e - 28 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 167, normalized size = 0.82 \[ \frac {2}{35} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 35 \, \sqrt {x e + d} a^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 132, normalized size = 0.65 \[ \frac {2 \sqrt {e x +d}\, \left (5 b^{3} e^{3} x^{3}+21 a \,b^{2} e^{3} x^{2}-6 b^{3} d \,e^{2} x^{2}+35 a^{2} b \,e^{3} x -28 a \,b^{2} d \,e^{2} x +8 b^{3} d^{2} e x +35 a^{3} e^{3}-70 a^{2} b d \,e^{2}+56 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{35 \left (b x +a \right )^{3} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 164, normalized size = 0.80 \[ \frac {2 \, {\left (5 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 56 \, a b^{2} d^{3} e - 70 \, a^{2} b d^{2} e^{2} + 35 \, a^{3} d e^{3} - {\left (b^{3} d e^{3} - 21 \, a b^{2} e^{4}\right )} x^{3} + {\left (2 \, b^{3} d^{2} e^{2} - 7 \, a b^{2} d e^{3} + 35 \, a^{2} b e^{4}\right )} x^{2} - {\left (8 \, b^{3} d^{3} e - 28 \, a b^{2} d^{2} e^{2} + 35 \, a^{2} b d e^{3} - 35 \, a^{3} e^{4}\right )} x\right )}}{35 \, \sqrt {e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 198, normalized size = 0.97 \[ \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^2\,x^4}{7}+\frac {2\,x^2\,\left (35\,a^2\,e^2-7\,a\,b\,d\,e+2\,b^2\,d^2\right )}{35\,e^2}-\frac {-70\,a^3\,d\,e^3+140\,a^2\,b\,d^2\,e^2-112\,a\,b^2\,d^3\,e+32\,b^3\,d^4}{35\,b\,e^4}+\frac {2\,b\,x^3\,\left (21\,a\,e-b\,d\right )}{35\,e}+\frac {x\,\left (70\,a^3\,e^4-70\,a^2\,b\,d\,e^3+56\,a\,b^2\,d^2\,e^2-16\,b^3\,d^3\,e\right )}{35\,b\,e^4}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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