Optimal. Leaf size=344 \[ \frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^4}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}+\frac {10 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]
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Rubi [A]
time = 0.15, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45}
\begin {gather*} -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 784
Rubi steps
\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^6} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^6} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {b^6}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^6}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^4}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^3}+\frac {10 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 315, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+60 b^5 (b d-a e) (d+e x)^5 \log (d+e x)\right )}{10 e^7 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(602\) vs.
\(2(263)=526\).
time = 0.08, size = 603, normalized size = 1.75
method | result | size |
risch | \(\frac {b^{6} x \sqrt {\left (b x +a \right )^{2}}}{e^{6} \left (b x +a \right )}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\left (-15 a^{2} b^{4} e^{5}+30 d \,e^{4} a \,b^{5}-15 b^{6} d^{2} e^{3}\right ) x^{4}-10 b^{3} e^{2} \left (a^{3} e^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x^{3}-5 b^{2} e \left (a^{4} e^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}\right ) x^{2}-\frac {b \left (3 a^{5} e^{5}+5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+30 a^{2} b^{3} d^{3} e^{2}-125 a \,b^{4} d^{4} e +77 b^{5} d^{5}\right ) x}{2}-\frac {2 e^{6} a^{6}+3 d \,e^{5} a^{5} b +5 d^{2} e^{4} a^{4} b^{2}+10 d^{3} e^{3} a^{3} b^{3}+30 d^{4} e^{2} a^{2} b^{4}-137 d^{5} e a \,b^{5}+87 d^{6} b^{6}}{10 e}\right )}{\left (b x +a \right ) e^{6} \left (e x +d \right )^{5}}+\frac {6 \sqrt {\left (b x +a \right )^{2}}\, b^{5} \left (a e -b d \right ) \ln \left (e x +d \right )}{\left (b x +a \right ) e^{7}}\) | \(388\) |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (-300 \ln \left (e x +d \right ) b^{6} d^{5} e x +60 \ln \left (e x +d \right ) a \,b^{5} e^{6} x^{5}-60 \ln \left (e x +d \right ) b^{6} d \,e^{5} x^{5}-15 a^{5} b \,e^{6} x -375 b^{6} d^{5} e x +50 b^{6} d \,e^{5} x^{5}-150 a^{2} b^{4} e^{6} x^{4}-50 b^{6} d^{2} e^{4} x^{4}-100 a^{3} b^{3} e^{6} x^{3}-400 b^{6} d^{3} e^{3} x^{3}-50 a^{4} b^{2} e^{6} x^{2}-600 b^{6} d^{4} e^{2} x^{2}-87 d^{6} b^{6}-2 e^{6} a^{6}+300 \ln \left (e x +d \right ) a \,b^{5} d \,e^{5} x^{4}+600 \ln \left (e x +d \right ) a \,b^{5} d^{2} e^{4} x^{3}+300 \ln \left (e x +d \right ) a \,b^{5} d^{4} e^{2} x +600 \ln \left (e x +d \right ) a \,b^{5} d^{3} e^{3} x^{2}+137 d^{5} e a \,b^{5}-30 d^{4} e^{2} a^{2} b^{4}-10 d^{3} e^{3} a^{3} b^{3}-5 d^{2} e^{4} a^{4} b^{2}-3 d \,e^{5} a^{5} b -60 \ln \left (e x +d \right ) b^{6} d^{6}+10 b^{6} e^{6} x^{6}+1100 a \,b^{5} d^{3} e^{3} x^{2}-25 a^{4} b^{2} d \,e^{5} x -50 a^{3} b^{3} d^{2} e^{4} x -150 a^{2} b^{4} d^{3} e^{3} x +625 a \,b^{5} d^{4} e^{2} x +60 \ln \left (e x +d \right ) a \,b^{5} d^{5} e +300 a \,b^{5} d \,e^{5} x^{4}-300 a^{2} b^{4} d \,e^{5} x^{3}+900 a \,b^{5} d^{2} e^{4} x^{3}-100 a^{3} b^{3} d \,e^{5} x^{2}-300 a^{2} b^{4} d^{2} e^{4} x^{2}-600 \ln \left (e x +d \right ) b^{6} d^{3} e^{3} x^{3}-300 \ln \left (e x +d \right ) b^{6} d^{2} e^{4} x^{4}-600 \ln \left (e x +d \right ) b^{6} d^{4} e^{2} x^{2}\right )}{10 \left (b x +a \right )^{5} e^{7} \left (e x +d \right )^{5}}\) | \(603\) |
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 = 2.27, size = 510, normalized size = 1.48 \begin {gather*} -\frac {87 \, b^{6} d^{6} - {\left (10 \, b^{6} x^{6} - 150 \, a^{2} b^{4} x^{4} - 100 \, a^{3} b^{3} x^{3} - 50 \, a^{4} b^{2} x^{2} - 15 \, a^{5} b x - 2 \, a^{6}\right )} e^{6} - {\left (50 \, b^{6} d x^{5} + 300 \, a b^{5} d x^{4} - 300 \, a^{2} b^{4} d x^{3} - 100 \, a^{3} b^{3} d x^{2} - 25 \, a^{4} b^{2} d x - 3 \, a^{5} b d\right )} e^{5} + 5 \, {\left (10 \, b^{6} d^{2} x^{4} - 180 \, a b^{5} d^{2} x^{3} + 60 \, a^{2} b^{4} d^{2} x^{2} + 10 \, a^{3} b^{3} d^{2} x + a^{4} b^{2} d^{2}\right )} e^{4} + 10 \, {\left (40 \, b^{6} d^{3} x^{3} - 110 \, a b^{5} d^{3} x^{2} + 15 \, a^{2} b^{4} d^{3} x + a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (120 \, b^{6} d^{4} x^{2} - 125 \, a b^{5} d^{4} x + 6 \, a^{2} b^{4} d^{4}\right )} e^{2} + {\left (375 \, b^{6} d^{5} x - 137 \, a b^{5} d^{5}\right )} e + 60 \, {\left (b^{6} d^{6} - a b^{5} x^{5} e^{6} + {\left (b^{6} d x^{5} - 5 \, a b^{5} d x^{4}\right )} e^{5} + 5 \, {\left (b^{6} d^{2} x^{4} - 2 \, a b^{5} d^{2} x^{3}\right )} e^{4} + 10 \, {\left (b^{6} d^{3} x^{3} - a b^{5} d^{3} x^{2}\right )} e^{3} + 5 \, {\left (2 \, b^{6} d^{4} x^{2} - a b^{5} d^{4} x\right )} e^{2} + {\left (5 \, b^{6} d^{5} x - a b^{5} d^{5}\right )} e\right )} \log \left (x e + d\right )}{10 \, {\left (x^{5} e^{12} + 5 \, d x^{4} e^{11} + 10 \, d^{2} x^{3} e^{10} + 10 \, d^{3} x^{2} e^{9} + 5 \, d^{4} x e^{8} + d^{5} e^{7}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 499, normalized size = 1.45 \begin {gather*} b^{6} x e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (b^{6} d \mathrm {sgn}\left (b x + a\right ) - a b^{5} e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (87 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 137 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 150 \, {\left (b^{6} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, a b^{5} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 22 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{3} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 5 \, {\left (77 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 125 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{10 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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