Optimal. Leaf size=344 \[ -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2} \]
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Rubi [A] time = 0.23, 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 = {770, 21, 43} \begin {gather*} -\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}-\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 {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 {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
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 = 0.16, 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 )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+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 )\right )}{10 e^7 (a+b x) (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.12, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 542, normalized size = 1.58 \begin {gather*} \frac {10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - a b^{5} d^{5} e + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \, {\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, 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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maple [B] time = 0.11, size = 603, normalized size = 1.75 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{5} e^{6} x^{5} \ln \left (e x +d \right )-60 b^{6} d \,e^{5} x^{5} \ln \left (e x +d \right )+10 b^{6} e^{6} x^{6}+300 a \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )-300 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+50 b^{6} d \,e^{5} x^{5}-150 a^{2} b^{4} e^{6} x^{4}+600 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+300 a \,b^{5} d \,e^{5} x^{4}-600 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-50 b^{6} d^{2} e^{4} x^{4}-100 a^{3} b^{3} e^{6} x^{3}-300 a^{2} b^{4} d \,e^{5} x^{3}+600 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+900 a \,b^{5} d^{2} e^{4} x^{3}-600 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-400 b^{6} d^{3} e^{3} x^{3}-50 a^{4} b^{2} e^{6} x^{2}-100 a^{3} b^{3} d \,e^{5} x^{2}-300 a^{2} b^{4} d^{2} e^{4} x^{2}+300 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )+1100 a \,b^{5} d^{3} e^{3} x^{2}-300 b^{6} d^{5} e x \ln \left (e x +d \right )-600 b^{6} d^{4} e^{2} x^{2}-15 a^{5} b \,e^{6} x -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 +60 a \,b^{5} d^{5} e \ln \left (e x +d \right )+625 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )-375 b^{6} d^{5} e x -2 a^{6} e^{6}-3 a^{5} b d \,e^{5}-5 a^{4} b^{2} d^{2} e^{4}-10 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}+137 a \,b^{5} d^{5} e -87 b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} \left (e x +d \right )^{5} e^{7}} \end {gather*}
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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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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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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