Optimal. Leaf size=314 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)} \]
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Rubi [A] time = 0.10, antiderivative size = 314, 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} \begin {gather*} \frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^6 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{3 e^6 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}{e^6 (a+b x)}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) \sqrt {d+e x}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{3 e^6 (a+b x) (d+e x)^{3/2}} \end {gather*}
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 )^{5/2}}{(d+e x)^{5/2}} \, 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)^{5/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{5/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{3/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {10 b^8 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{3/2}}{e^5}+\frac {b^{10} (d+e x)^{5/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}-\frac {2 b^4 (b d-a e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 235, normalized size = 0.75 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (7 a^5 e^5+35 a^4 b e^4 (2 d+3 e x)-70 a^3 b^2 e^3 \left (8 d^2+12 d e x+3 e^2 x^2\right )+70 a^2 b^3 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b^4 e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 30.68, size = 343, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-7 a^5 e^5-105 a^4 b e^4 (d+e x)+35 a^4 b d e^4-70 a^3 b^2 d^2 e^3+210 a^3 b^2 e^3 (d+e x)^2+420 a^3 b^2 d e^3 (d+e x)+70 a^2 b^3 d^3 e^2-630 a^2 b^3 d^2 e^2 (d+e x)+70 a^2 b^3 e^2 (d+e x)^3-630 a^2 b^3 d e^2 (d+e x)^2-35 a b^4 d^4 e+420 a b^4 d^3 e (d+e x)+630 a b^4 d^2 e (d+e x)^2+21 a b^4 e (d+e x)^4-140 a b^4 d e (d+e x)^3+7 b^5 d^5-105 b^5 d^4 (d+e x)-210 b^5 d^3 (d+e x)^2+70 b^5 d^2 (d+e x)^3+3 b^5 (d+e x)^5-21 b^5 d (d+e x)^4\right )}{21 e^5 (d+e x)^{3/2} (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 283, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 460, normalized size = 1.46 \begin {gather*} \frac {2}{21} \, {\left (3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} e^{36} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d e^{36} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{36} \mathrm {sgn}\left (b x + a\right ) - 210 \, \sqrt {x e + d} b^{5} d^{3} e^{36} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} e^{37} \mathrm {sgn}\left (b x + a\right ) - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d e^{37} \mathrm {sgn}\left (b x + a\right ) + 630 \, \sqrt {x e + d} a b^{4} d^{2} e^{37} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{38} \mathrm {sgn}\left (b x + a\right ) - 630 \, \sqrt {x e + d} a^{2} b^{3} d e^{38} \mathrm {sgn}\left (b x + a\right ) + 210 \, \sqrt {x e + d} a^{3} b^{2} e^{39} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, {\left (x e + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 90 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 60 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, {\left (x e + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 289, normalized size = 0.92 \begin {gather*} -\frac {2 \left (-3 b^{5} e^{5} x^{5}-21 a \,b^{4} e^{5} x^{4}+6 b^{5} d \,e^{4} x^{4}-70 a^{2} b^{3} e^{5} x^{3}+56 a \,b^{4} d \,e^{4} x^{3}-16 b^{5} d^{2} e^{3} x^{3}-210 a^{3} b^{2} e^{5} x^{2}+420 a^{2} b^{3} d \,e^{4} x^{2}-336 a \,b^{4} d^{2} e^{3} x^{2}+96 b^{5} d^{3} e^{2} x^{2}+105 a^{4} b \,e^{5} x -840 a^{3} b^{2} d \,e^{4} x +1680 a^{2} b^{3} d^{2} e^{3} x -1344 a \,b^{4} d^{3} e^{2} x +384 b^{5} d^{4} e x +7 a^{5} e^{5}+70 a^{4} b d \,e^{4}-560 a^{3} b^{2} d^{2} e^{3}+1120 a^{2} b^{3} d^{3} e^{2}-896 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 272, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{21 \, {\left (e^{7} x + d e^{6}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.57, size = 330, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^4\,x^5}{7\,e^2}-\frac {14\,a^5\,e^5+140\,a^4\,b\,d\,e^4-1120\,a^3\,b^2\,d^2\,e^3+2240\,a^2\,b^3\,d^3\,e^2-1792\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{21\,b\,e^7}+\frac {2\,b^3\,x^4\,\left (7\,a\,e-2\,b\,d\right )}{7\,e^3}-\frac {x\,\left (210\,a^4\,b\,e^5-1680\,a^3\,b^2\,d\,e^4+3360\,a^2\,b^3\,d^2\,e^3-2688\,a\,b^4\,d^3\,e^2+768\,b^5\,d^4\,e\right )}{21\,b\,e^7}+\frac {4\,b^2\,x^3\,\left (35\,a^2\,e^2-28\,a\,b\,d\,e+8\,b^2\,d^2\right )}{21\,e^4}+\frac {4\,b\,x^2\,\left (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 )}{7\,e^5}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (21\,a\,e^7+21\,b\,d\,e^6\right )\,\sqrt {d+e\,x}}{21\,b\,e^7}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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