Optimal. Leaf size=209 \[ -\frac {4 e^3 (b d-a e)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} -\frac {4 e^3 (b d-a e)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^4}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {(b d-a e)^4}{b^9 (a+b x)^5}+\frac {4 e (b d-a e)^3}{b^9 (a+b x)^4}+\frac {6 e^2 (b d-a e)^2}{b^9 (a+b x)^3}+\frac {4 e^3 (b d-a e)}{b^9 (a+b x)^2}+\frac {e^4}{b^9 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 e^3 (b d-a e)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4}{4 b^5 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 e (b d-a e)^3}{3 b^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^2 (b d-a e)^2}{b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 138, normalized size = 0.66 \begin {gather*} \frac {-\left ((b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )\right )+12 e^4 (a+b x)^4 \log (a+b x)}{12 b^5 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 267, normalized size = 1.28
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {4 e^{3} \left (a e -b d \right ) x^{3}}{b^{2}}+\frac {3 e^{2} \left (3 a^{2} e^{2}-2 a b d e -b^{2} d^{2}\right ) x^{2}}{b^{3}}+\frac {2 e \left (11 e^{3} a^{3}-6 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) x}{3 b^{4}}+\frac {25 e^{4} a^{4}-12 a^{3} b d \,e^{3}-6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e -3 b^{4} d^{4}}{12 b^{5}}\right )}{\left (b x +a \right )^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{4} \ln \left (b x +a \right )}{\left (b x +a \right ) b^{5}}\) | \(202\) |
default | \(\frac {\left (12 \ln \left (b x +a \right ) b^{4} e^{4} x^{4}+48 \ln \left (b x +a \right ) a \,b^{3} e^{4} x^{3}+72 \ln \left (b x +a \right ) a^{2} b^{2} e^{4} x^{2}+48 a \,b^{3} e^{4} x^{3}-48 b^{4} d \,e^{3} x^{3}+48 \ln \left (b x +a \right ) a^{3} b \,e^{4} x +108 a^{2} b^{2} e^{4} x^{2}-72 a \,b^{3} d \,e^{3} x^{2}-36 b^{4} d^{2} e^{2} x^{2}+12 \ln \left (b x +a \right ) a^{4} e^{4}+88 a^{3} b \,e^{4} x -48 a^{2} b^{2} d \,e^{3} x -24 a \,b^{3} d^{2} e^{2} x -16 b^{4} d^{3} e x +25 e^{4} a^{4}-12 a^{3} b d \,e^{3}-6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e -3 b^{4} d^{4}\right ) \left (b x +a \right )}{12 b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(267\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs.
\(2 (152) = 304\).
time = 0.29, size = 321, normalized size = 1.54 \begin {gather*} -\frac {1}{3} \, d^{3} {\left (\frac {4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {3 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} e - \frac {1}{2} \, d^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} e^{2} - \frac {1}{3} \, d {\left (\frac {12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}} + \frac {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\left (x + \frac {a}{b}\right )}^{4}}\right )} e^{3} + \frac {1}{12} \, {\left (\frac {48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac {12 \, \log \left (b x + a\right )}{b^{5}}\right )} e^{4} - \frac {d^{4}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.52, size = 240, normalized size = 1.15 \begin {gather*} -\frac {3 \, b^{4} d^{4} - 12 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} e^{4} \log \left (b x + a\right ) - {\left (48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}\right )} e^{4} + 12 \, {\left (4 \, b^{4} d x^{3} + 6 \, a b^{3} d x^{2} + 4 \, a^{2} b^{2} d x + a^{3} b d\right )} e^{3} + 6 \, {\left (6 \, b^{4} d^{2} x^{2} + 4 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2}\right )} e^{2} + 4 \, {\left (4 \, b^{4} d^{3} x + a b^{3} d^{3}\right )} e}{12 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\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 (d + e x\right )^{4}}{\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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Giac [A]
time = 0.98, size = 190, normalized size = 0.91 \begin {gather*} \frac {e^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {48 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} x^{3} + 36 \, {\left (b^{3} d^{2} e^{2} + 2 \, a b^{2} d e^{3} - 3 \, a^{2} b e^{4}\right )} x^{2} + 8 \, {\left (2 \, b^{3} d^{3} e + 3 \, a b^{2} d^{2} e^{2} + 6 \, a^{2} b d e^{3} - 11 \, a^{3} e^{4}\right )} x + \frac {3 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 25 \, a^{4} e^{4}}{b}}{12 \, {\left (b x + a\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right )} \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 (d+e\,x\right )}^4}{{\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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