Optimal. Leaf size=200 \[ \frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4} \]
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Rubi [A] time = 0.09, 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 = {646, 43} \[ -\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}+\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^5}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^4 (a+b x) (d+e x)^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^4 (a+b x) (d+e x)^7} \]
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)^8} \, 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)^8} \, 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)^8}+\frac {3 b^4 (b d-a e)^2}{e^3 (d+e x)^7}-\frac {3 b^5 (b d-a e)}{e^3 (d+e x)^6}+\frac {b^6}{e^3 (d+e x)^5}\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}}{7 e^4 (a+b x) (d+e x)^7}-\frac {b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^4 (a+b x) (d+e x)^6}+\frac {3 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}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^4 (a+b x) (d+e x)^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 112, normalized size = 0.56 \[ -\frac {\sqrt {(a+b x)^2} \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{140 e^4 (a+b x) (d+e x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 182, normalized size = 0.91 \[ -\frac {35 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 4 \, a b^{2} d^{2} e + 10 \, a^{2} b d e^{2} + 20 \, a^{3} e^{3} + 21 \, {\left (b^{3} d e^{2} + 4 \, a b^{2} e^{3}\right )} x^{2} + 7 \, {\left (b^{3} d^{2} e + 4 \, a b^{2} d e^{2} + 10 \, a^{2} b e^{3}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 169, normalized size = 0.84 \[ -\frac {{\left (35 \, b^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{3} d x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{3} d^{2} x e \mathrm {sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, a b^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, a b^{2} d x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{2} b x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{140 \, {\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 131, normalized size = 0.66 \[ -\frac {\left (35 b^{3} e^{3} x^{3}+84 a \,b^{2} e^{3} x^{2}+21 b^{3} d \,e^{2} x^{2}+70 a^{2} b \,e^{3} x +28 a \,b^{2} d \,e^{2} x +7 b^{3} d^{2} e x +20 a^{3} e^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{140 \left (e x +d \right )^{7} \left (b x +a \right )^{3} e^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 284, normalized size = 1.42 \[ \frac {\left (\frac {2\,b^3\,d-3\,a\,b^2\,e}{5\,e^4}+\frac {b^3\,d}{5\,e^4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {3\,a^2\,b\,e^2-3\,a\,b^2\,d\,e+b^3\,d^2}{6\,e^4}+\frac {d\,\left (\frac {b^3\,d}{6\,e^3}-\frac {b^2\,\left (3\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^3}{7\,e}-\frac {d\,\left (\frac {3\,a^2\,b}{7\,e}-\frac {d\,\left (\frac {3\,a\,b^2}{7\,e}-\frac {b^3\,d}{7\,e^2}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^4\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \]
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 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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