Optimal. Leaf size=252 \[ -\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {10 e^3 (a+b x) (b d-a e)^2 \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {20 e^2 (b d-a e)^3}{3 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{6 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 x (a+b x) (4 b d-3 a e)}{3 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^5 x^2 (a+b x)}{6 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.19, antiderivative size = 252, 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 = {768, 646, 43} \begin {gather*} \frac {5 e^4 x (a+b x) (4 b d-3 a e)}{3 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {20 e^2 (b d-a e)^3}{3 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 e^3 (a+b x) (b d-a e)^2 \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{6 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {5 e^5 x^2 (a+b x)}{6 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 768
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=-\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {(5 e) \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {\left (5 b e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^4}{\left (a b+b^2 x\right )^3} \, dx}{3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}+\frac {\left (5 b e \left (a b+b^2 x\right )\right ) \int \left (\frac {e^3 (4 b d-3 a e)}{b^7}+\frac {e^4 x}{b^6}+\frac {(b d-a e)^4}{b^7 (a+b x)^3}+\frac {4 e (b d-a e)^3}{b^7 (a+b x)^2}+\frac {6 e^2 (b d-a e)^2}{b^7 (a+b x)}\right ) \, dx}{3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^5}{3 b \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac {20 e^2 (b d-a e)^3}{3 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^4}{6 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (4 b d-3 a e) x (a+b x)}{3 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^5 x^2 (a+b x)}{6 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 e^3 (b d-a e)^2 (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 232, normalized size = 0.92 \begin {gather*} \frac {47 a^5 e^5+a^4 b e^4 (81 e x-130 d)+a^3 b^2 e^3 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 b^3 e^2 \left (20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right )-5 a b^4 e \left (d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right )+60 e^3 (a+b x)^3 (b d-a e)^2 \log (a+b x)+b^5 \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )}{6 b^6 \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 8.17, size = 6538, normalized size = 25.94 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 426, normalized size = 1.69 \begin {gather*} \frac {3 \, b^{5} e^{5} x^{5} - 2 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 20 \, a^{2} b^{3} d^{3} e^{2} + 110 \, a^{3} b^{2} d^{2} e^{3} - 130 \, a^{4} b d e^{4} + 47 \, a^{5} e^{5} + 15 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 9 \, {\left (10 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (20 \, b^{5} d^{3} e^{2} - 60 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 3 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (5 \, b^{5} d^{4} e + 20 \, a b^{4} d^{3} e^{2} - 90 \, a^{2} b^{3} d^{2} e^{3} + 90 \, a^{3} b^{2} d e^{4} - 27 \, a^{4} b e^{5}\right )} x + 60 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{5}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 495, normalized size = 1.96 \begin {gather*} \frac {\left (3 b^{5} e^{5} x^{5}+60 a^{2} b^{3} e^{5} x^{3} \ln \left (b x +a \right )-120 a \,b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )-15 a \,b^{4} e^{5} x^{4}+60 b^{5} d^{2} e^{3} x^{3} \ln \left (b x +a \right )+30 b^{5} d \,e^{4} x^{4}+180 a^{3} b^{2} e^{5} x^{2} \ln \left (b x +a \right )-360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )-63 a^{2} b^{3} e^{5} x^{3}+180 a \,b^{4} d^{2} e^{3} x^{2} \ln \left (b x +a \right )+90 a \,b^{4} d \,e^{4} x^{3}+180 a^{4} b \,e^{5} x \ln \left (b x +a \right )-360 a^{3} b^{2} d \,e^{4} x \ln \left (b x +a \right )-9 a^{3} b^{2} e^{5} x^{2}+180 a^{2} b^{3} d^{2} e^{3} x \ln \left (b x +a \right )-90 a^{2} b^{3} d \,e^{4} x^{2}+180 a \,b^{4} d^{2} e^{3} x^{2}-60 b^{5} d^{3} e^{2} x^{2}+60 a^{5} e^{5} \ln \left (b x +a \right )-120 a^{4} b d \,e^{4} \ln \left (b x +a \right )+81 a^{4} b \,e^{5} x +60 a^{3} b^{2} d^{2} e^{3} \ln \left (b x +a \right )-270 a^{3} b^{2} d \,e^{4} x +270 a^{2} b^{3} d^{2} e^{3} x -60 a \,b^{4} d^{3} e^{2} x -15 b^{5} d^{4} e x +47 a^{5} e^{5}-130 a^{4} b d \,e^{4}+110 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e -2 b^{5} d^{5}\right ) \left (b x +a \right )^{2}}{6 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 1010, normalized size = 4.01
result too large to display
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 (d+e\,x\right )}^5}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,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 (d + e x\right )^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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