Optimal. Leaf size=200 \[ \frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x) (d+e x)^8}-\frac {3 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x) (d+e x)^7}+\frac {b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^6}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^4 (a+b x) (d+e x)^6}-\frac {3 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}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{8 e^4 (a+b x) (d+e x)^8}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^9} \, 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)^9} \, 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)^9}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^8}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^7}+\frac {b^6}{e^3 (d+e x)^6}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^4 (a+b x) (d+e x)^8}-\frac {3 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}+\frac {b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^6}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 112, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )}{280 e^4 (a+b x) (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 131, normalized size = 0.66
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{3} x^{3}}{5 e}-\frac {b^{2} \left (5 a e +b d \right ) x^{2}}{10 e^{2}}-\frac {b \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x}{35 e^{3}}-\frac {35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}}{280 e^{4}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{8}}\) | \(126\) |
gosper | \(-\frac {\left (56 b^{3} e^{3} x^{3}+140 a \,b^{2} e^{3} x^{2}+28 b^{3} d \,e^{2} x^{2}+120 a^{2} b \,e^{3} x +40 a \,b^{2} d \,e^{2} x +8 b^{3} d^{2} e x +35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 e^{4} \left (e x +d \right )^{8} \left (b x +a \right )^{3}}\) | \(131\) |
default | \(-\frac {\left (56 b^{3} e^{3} x^{3}+140 a \,b^{2} e^{3} x^{2}+28 b^{3} d \,e^{2} x^{2}+120 a^{2} b \,e^{3} x +40 a \,b^{2} d \,e^{2} x +8 b^{3} d^{2} e x +35 e^{3} a^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{280 e^{4} \left (e x +d \right )^{8} \left (b x +a \right )^{3}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 174, normalized size = 0.87 \begin {gather*} -\frac {b^{3} d^{3} + {\left (56 \, b^{3} x^{3} + 140 \, a b^{2} x^{2} + 120 \, a^{2} b x + 35 \, a^{3}\right )} e^{3} + {\left (28 \, b^{3} d x^{2} + 40 \, a b^{2} d x + 15 \, a^{2} b d\right )} e^{2} + {\left (8 \, b^{3} d^{2} x + 5 \, a b^{2} d^{2}\right )} e}{280 \, {\left (x^{8} e^{12} + 8 \, d x^{7} e^{11} + 28 \, d^{2} x^{6} e^{10} + 56 \, d^{3} x^{5} e^{9} + 70 \, d^{4} x^{4} e^{8} + 56 \, d^{5} x^{3} e^{7} + 28 \, d^{6} x^{2} e^{6} + 8 \, d^{7} x e^{5} + d^{8} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.47, size = 169, normalized size = 0.84 \begin {gather*} -\frac {{\left (56 \, b^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{3} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{3} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 140 \, a b^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 40 \, a b^{2} d x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{2} b x e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, 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 )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 284, normalized size = 1.42 \begin {gather*} \frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{6\,e^4}+\frac {b^3\,d}{6\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{7\,e^4}+\frac {d\,\left (\frac {b^3\,d}{7\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {a^3}{8\,e}-\frac {d\,\left (\frac {3\,a^2\,b}{8\,e}-\frac {d\,\left (\frac {3\,a\,b^2}{8\,e}-\frac {b^3\,d}{8\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^4\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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