Optimal. Leaf size=345 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)}+\frac {15 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)} \]
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Rubi [A] time = 0.30, antiderivative size = 345, 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 {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)}+\frac {15 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (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)^2} \, 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)^2} \, 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)^2} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {15 b^2 (b d-a e)^4}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^2}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^3}{e^6}+\frac {b^6 (d+e x)^4}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {15 b^2 (b d-a e)^4 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}}{e^7 (a+b x) (d+e x)}-\frac {10 b^3 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {3 b^5 (b d-a e) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac {b^6 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {6 b (b d-a e)^5 \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 = 0.20, size = 320, normalized size = 0.93 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-10 a^6 e^6+60 a^5 b d e^5+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 496, normalized size = 1.44 \begin {gather*} \frac {2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \, {\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \, {\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, 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} - a^{5} b d e^{5} + {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, 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} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{8} x + d 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 = 519, normalized size = 1.50 \begin {gather*} -6 \, {\left (b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{10} \, {\left (2 \, b^{6} x^{5} e^{8} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{6} d x^{4} e^{7} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{2} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 50 \, b^{6} d^{4} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{5} x^{4} e^{8} \mathrm {sgn}\left (b x + a\right ) - 40 \, a b^{5} d x^{3} e^{7} \mathrm {sgn}\left (b x + a\right ) + 90 \, a b^{5} d^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - 240 \, a b^{5} d^{3} x e^{5} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} x^{3} e^{8} \mathrm {sgn}\left (b x + a\right ) - 150 \, a^{2} b^{4} d x^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x e^{6} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} x^{2} e^{8} \mathrm {sgn}\left (b x + a\right ) - 400 \, a^{3} b^{3} d x e^{7} \mathrm {sgn}\left (b x + a\right ) + 150 \, a^{4} b^{2} x e^{8} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} - \frac {{\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{x e + d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 601, normalized size = 1.74 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{6} e^{6} x^{6}+15 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+50 a^{2} b^{4} e^{6} x^{4}-25 a \,b^{5} d \,e^{5} x^{4}+5 b^{6} d^{2} e^{4} x^{4}+100 a^{3} b^{3} e^{6} x^{3}-100 a^{2} b^{4} d \,e^{5} x^{3}+50 a \,b^{5} d^{2} e^{4} x^{3}-10 b^{6} d^{3} e^{3} x^{3}+60 a^{5} b \,e^{6} x \ln \left (e x +d \right )-300 a^{4} b^{2} d \,e^{5} x \ln \left (e x +d \right )+150 a^{4} b^{2} e^{6} x^{2}+600 a^{3} b^{3} d^{2} e^{4} x \ln \left (e x +d \right )-300 a^{3} b^{3} d \,e^{5} x^{2}-600 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+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 )-150 a \,b^{5} d^{3} e^{3} x^{2}-60 b^{6} d^{5} e x \ln \left (e x +d \right )+30 b^{6} d^{4} e^{2} x^{2}+60 a^{5} b d \,e^{5} \ln \left (e x +d \right )-300 a^{4} b^{2} d^{2} e^{4} \ln \left (e x +d \right )+150 a^{4} b^{2} d \,e^{5} x +600 a^{3} b^{3} d^{3} e^{3} \ln \left (e x +d \right )-400 a^{3} b^{3} d^{2} e^{4} x -600 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )+450 a^{2} b^{4} d^{3} e^{3} x +300 a \,b^{5} d^{5} e \ln \left (e x +d \right )-240 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )+50 b^{6} d^{5} e x -10 a^{6} e^{6}+60 a^{5} b d \,e^{5}-150 a^{4} b^{2} d^{2} e^{4}+200 a^{3} b^{3} d^{3} e^{3}-150 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -10 b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} \left (e x +d \right ) 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 )}^2} \,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 )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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