Optimal. Leaf size=149 \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {646, 45, 37} \[ \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(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 )^5}{(d+e x)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^8} \, dx}{4 b^3 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^7} \, dx}{28 b^2 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{168 (b d-a e)^3 (d+e x)^6}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 223, normalized size = 1.50 \[ -\frac {\sqrt {(a+b x)^2} \left (21 a^5 e^5+15 a^4 b e^4 (d+8 e x)+10 a^3 b^2 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a^2 b^3 e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a b^4 e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+b^5 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{168 e^6 (a+b x) (d+e x)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 337, normalized size = 2.26 \[ -\frac {56 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 3 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} + 21 \, a^{5} e^{5} + 70 \, {\left (b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 56 \, {\left (b^{5} d^{2} e^{3} + 3 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} + 28 \, {\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 8 \, {\left (b^{5} d^{4} e + 3 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 15 \, a^{4} b e^{5}\right )} x}{168 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 381, normalized size = 2.56 \[ -\frac {{\left (56 \, b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 28 \, b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 168 \, a b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, a b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 24 \, a b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 168 \, a^{2} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 48 \, a^{2} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 80 \, a^{3} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 120 \, a^{4} b x e^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{168 \, {\left (x e + d\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 288, normalized size = 1.93 \[ -\frac {\left (56 b^{5} e^{5} x^{5}+210 a \,b^{4} e^{5} x^{4}+70 b^{5} d \,e^{4} x^{4}+336 a^{2} b^{3} e^{5} x^{3}+168 a \,b^{4} d \,e^{4} x^{3}+56 b^{5} d^{2} e^{3} x^{3}+280 a^{3} b^{2} e^{5} x^{2}+168 a^{2} b^{3} d \,e^{4} x^{2}+84 a \,b^{4} d^{2} e^{3} x^{2}+28 b^{5} d^{3} e^{2} x^{2}+120 a^{4} b \,e^{5} x +80 a^{3} b^{2} d \,e^{4} x +48 a^{2} b^{3} d^{2} e^{3} x +24 a \,b^{4} d^{3} e^{2} x +8 b^{5} d^{4} e x +21 a^{5} e^{5}+15 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+6 a^{2} b^{3} d^{3} e^{2}+3 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{168 \left (e x +d \right )^{8} \left (b x +a \right )^{5} e^{6}} \]
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 = 0.74, size = 687, normalized size = 4.61 \[ \frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{4\,e^6}+\frac {b^5\,d}{4\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{7\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{7\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{7\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{7\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{7\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{5\,e^6}+\frac {d\,\left (\frac {b^5\,d}{5\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{5\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {a^5}{8\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{8\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{8\,e}-\frac {b^5\,d}{8\,e^2}\right )}{e}-\frac {5\,a^2\,b^3}{4\,e}\right )}{e}+\frac {5\,a^3\,b^2}{4\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{6\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{6\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{6\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{6\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \]
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 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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