Optimal. Leaf size=208 \[ \frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}} \]
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Rubi [A] time = 0.07, 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 = {646, 43} \[ -\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac {6 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}} \]
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}}{(d+e x)^{11/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{(d+e x)^{11/2}} \, 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}{e^3 (d+e x)^{11/2}}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^{9/2}}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^{7/2}}+\frac {b^6}{e^3 (d+e x)^{5/2}}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac {6 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac {6 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 120, normalized size = 0.58 \[ -\frac {2 \sqrt {(a+b x)^2} \left (35 a^3 e^3+15 a^2 b e^2 (2 d+9 e x)+3 a b^2 e \left (8 d^2+36 d e x+63 e^2 x^2\right )+b^3 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )\right )}{315 e^4 (a+b x) (d+e x)^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 170, normalized size = 0.82 \[ -\frac {2 \, {\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \, {\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 194, normalized size = 0.93 \[ -\frac {2 \, {\left (105 \, {\left (x e + d\right )}^{3} b^{3} \mathrm {sgn}\left (b x + a\right ) - 189 \, {\left (x e + d\right )}^{2} b^{3} d \mathrm {sgn}\left (b x + a\right ) + 135 \, {\left (x e + d\right )} b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 35 \, b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 189 \, {\left (x e + d\right )}^{2} a b^{2} e \mathrm {sgn}\left (b x + a\right ) - 270 \, {\left (x e + d\right )} a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + 105 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 135 \, {\left (x e + d\right )} a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right ) - 105 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{315 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 132, normalized size = 0.63 \[ -\frac {2 \left (105 b^{3} e^{3} x^{3}+189 a \,b^{2} e^{3} x^{2}+126 b^{3} d \,e^{2} x^{2}+135 a^{2} b \,e^{3} x +108 a \,b^{2} d \,e^{2} x +72 b^{3} d^{2} e x +35 a^{3} e^{3}+30 a^{2} b d \,e^{2}+24 a \,b^{2} d^{2} e +16 b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {9}{2}} \left (b x +a \right )^{3} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 159, normalized size = 0.76 \[ -\frac {2 \, {\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \, {\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt {e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 268, normalized size = 1.29 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {\frac {2\,a^3\,e^3}{9}+\frac {4\,a^2\,b\,d\,e^2}{21}+\frac {16\,a\,b^2\,d^2\,e}{105}+\frac {32\,b^3\,d^3}{315}}{b\,e^8}+\frac {2\,x\,\left (15\,a^2\,e^2+12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{35\,e^7}+\frac {2\,b^2\,x^3}{3\,e^5}+\frac {2\,b\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{5\,e^6}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (a\,e^8+4\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{b\,e^8}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {d^2\,x^2\,\left (6\,a\,e+4\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]
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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