Optimal. Leaf size=223 \[ -\frac {b^2}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {2 b e (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac {e (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac {3 b^2 e (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {3 b^2 e (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
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Rubi [A] time = 0.15, antiderivative size = 223, 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, 44} \begin {gather*} -\frac {b^2}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {2 b e (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac {e (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac {3 b^2 e (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac {3 b^2 e (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 44
Rule 770
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {a+b x}{\left (a b+b^2 x\right )^3 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{(a+b x)^2 (d+e x)^3} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)^2}-\frac {3 b^3 e}{(b d-a e)^4 (a+b x)}+\frac {e^2}{(b d-a e)^2 (d+e x)^3}+\frac {2 b e^2}{(b d-a e)^3 (d+e x)^2}+\frac {3 b^2 e^2}{(b d-a e)^4 (d+e x)}\right ) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {b^2}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (a+b x)}{2 (b d-a e)^2 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b e (a+b x)}{(b d-a e)^3 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b^2 e (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2 e (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 135, normalized size = 0.61 \begin {gather*} \frac {-(b d-a e) \left (-a^2 e^2+a b e (5 d+3 e x)+b^2 \left (2 d^2+9 d e x+6 e^2 x^2\right )\right )-6 b^2 e (a+b x) (d+e x)^2 \log (a+b x)+6 b^2 e (a+b x) (d+e x)^2 \log (d+e x)}{2 \sqrt {(a+b x)^2} (d+e x)^2 (b d-a e)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 53.00, size = 5578, normalized size = 25.01 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 495, normalized size = 2.22 \begin {gather*} -\frac {2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e + {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e + {\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + {\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} + {\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} + {\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} + {\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.85, size = 461, normalized size = 2.07 \begin {gather*} -\frac {3 \, a b^{2} e \log \left ({\left | b + \frac {a}{x} \right |}\right )}{a b^{4} d^{4} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) - 4 \, a^{2} b^{3} d^{3} e \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) + 6 \, a^{3} b^{2} d^{2} e^{2} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) - 4 \, a^{4} b d e^{3} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) + a^{5} e^{4} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right )} + \frac {3 \, b^{2} d e \log \left ({\left | \frac {d}{x} + e \right |}\right )}{b^{4} d^{5} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) - 4 \, a b^{3} d^{4} e \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) + 6 \, a^{2} b^{2} d^{3} e^{2} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) - 4 \, a^{3} b d^{2} e^{3} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right ) + a^{4} d e^{4} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right )} + \frac {2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} - 6 \, a^{2} b^{2} d e^{4} + a^{3} b e^{5} + \frac {4 \, b^{4} d^{4} e + 2 \, a b^{3} d^{3} e^{2} - 3 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}}{x} + \frac {2 \, {\left (b^{4} d^{5} - a b^{3} d^{4} e + 3 \, a^{2} b^{2} d^{3} e^{2} - 4 \, a^{3} b d^{2} e^{3} + a^{4} d e^{4}\right )}}{x^{2}}}{2 \, {\left (b d - a e\right )}^{4} a {\left (b + \frac {a}{x}\right )} d^{2} {\left (\frac {d}{x} + e\right )}^{2} \mathrm {sgn}\left (\frac {b}{x} + \frac {a}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 331, normalized size = 1.48 \begin {gather*} -\frac {\left (6 b^{3} e^{3} x^{3} \ln \left (b x +a \right )-6 b^{3} e^{3} x^{3} \ln \left (e x +d \right )+6 a \,b^{2} e^{3} x^{2} \ln \left (b x +a \right )-6 a \,b^{2} e^{3} x^{2} \ln \left (e x +d \right )+12 b^{3} d \,e^{2} x^{2} \ln \left (b x +a \right )-12 b^{3} d \,e^{2} x^{2} \ln \left (e x +d \right )+12 a \,b^{2} d \,e^{2} x \ln \left (b x +a \right )-12 a \,b^{2} d \,e^{2} x \ln \left (e x +d \right )-6 a \,b^{2} e^{3} x^{2}+6 b^{3} d^{2} e x \ln \left (b x +a \right )-6 b^{3} d^{2} e x \ln \left (e x +d \right )+6 b^{3} d \,e^{2} x^{2}-3 a^{2} b \,e^{3} x +6 a \,b^{2} d^{2} e \ln \left (b x +a \right )-6 a \,b^{2} d^{2} e \ln \left (e x +d \right )-6 a \,b^{2} d \,e^{2} x +9 b^{3} d^{2} e x +a^{3} e^{3}-6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +2 b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{2 \left (e x +d \right )^{2} \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}} \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 {a+b\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,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 {a + b x}{\left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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