Optimal. Leaf size=106 \[ -\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (a+b x) (b d-a e) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}{b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {768, 640, 608, 31} \begin {gather*} -\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (a+b x) (b d-a e) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 608
Rule 640
Rule 768
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(2 e) \int \frac {d+e x}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{b}\\ &=-\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}{b^3}+\frac {(2 e (b d-a e)) \int \frac {1}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx}{b^2}\\ &=-\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}{b^3}+\frac {\left (2 e (b d-a e) \left (a b+b^2 x\right )\right ) \int \frac {1}{a b+b^2 x} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^2}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2}}{b^3}+\frac {2 e (b d-a e) (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 0.70 \begin {gather*} \frac {-a^2 e^2+a b e (2 d+e x)-2 e (a+b x) (a e-b d) \log (a+b x)+b^2 \left (e^2 x^2-d^2\right )}{b^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.37, size = 924, normalized size = 8.72 \begin {gather*} \frac {a \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) e^2}{\left (b^2\right )^{3/2}}+\frac {a \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) e^2}{b^3}+\frac {-2 e^2 x^3 b^4+2 d^2 x b^4-2 a d^2 b^3-3 a e^2 x^2 b^3+a^2 e^2 x b^2-2 a^3 e^2 b+\sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-2 b^2 d^2-2 a^2 e^2+2 b^2 e^2 x^2+a b e^2 x\right )}{2 b^4 \sqrt {b^2} x (a+b x)-2 b^5 x \sqrt {a^2+2 b x a+b^2 x^2}}+\left (\frac {a e^2}{\left (b^2\right )^{3/2}}-\frac {a e^2}{b^3}\right ) \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )+\frac {-\frac {2 d e x^2 \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b^3}{\left (b^2\right )^{3/2}}-\frac {2 d e x^2 \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b^3}{\left (b^2\right )^{3/2}}+\frac {4 a^2 d e b}{\left (b^2\right )^{3/2}}-\frac {4 a d e x}{\sqrt {b^2}}+4 d e x^2 \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right )-\frac {4 d e x \sqrt {a^2+2 b x a+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right )}{\sqrt {b^2}}-\frac {2 a d e x \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{\sqrt {b^2}}-\frac {2 a d e x \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{\sqrt {b^2}}+\frac {4 a d e x \tanh ^{-1}\left (\frac {\sqrt {a^2+2 b x a+b^2 x^2}-\sqrt {b^2} x}{a}\right )}{b}+\frac {2 d e x \sqrt {a^2+2 b x a+b^2 x^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{b}+\frac {2 d e x \sqrt {a^2+2 b x a+b^2 x^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )}{b}+\frac {4 a d e \sqrt {a^2+2 b x a+b^2 x^2}}{b^2}}{\left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 92, normalized size = 0.87 \begin {gather*} \frac {b^{2} e^{2} x^{2} + a b e^{2} x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \, {\left (a b d e - a^{2} e^{2} + {\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 148, normalized size = 1.40 \begin {gather*} -\frac {2 \, {\left (b d e - a e^{2}\right )} \log \left ({\left | -3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{2} a b - a^{3} b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{3} {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}}\right )} a^{2} {\left | b \right |} \right |}\right )}{3 \, b^{2} {\left | b \right |}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} e^{2}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 116, normalized size = 1.09 \begin {gather*} -\frac {\left (2 a b \,e^{2} x \ln \left (b x +a \right )-2 b^{2} d e x \ln \left (b x +a \right )-b^{2} e^{2} x^{2}+2 a^{2} e^{2} \ln \left (b x +a \right )-2 a b d e \ln \left (b x +a \right )-a b \,e^{2} x +a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (b x +a \right )^{2}}{\left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 272, normalized size = 2.57 \begin {gather*} \frac {e^{2} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b} - \frac {3 \, a e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, a^{2} e^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} - \frac {6 \, a^{2} e^{2} x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (2 \, b d e + a e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {b d^{2} + 2 \, a d e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {a d^{2}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, a^{3} e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, {\left (2 \, b d e + a e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, b d e + a e^{2}\right )} a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (b d^{2} + 2 \, a d e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}{{\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 {\left (a + b x\right ) \left (d + e x\right )^{2}}{\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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