Optimal. Leaf size=125 \[ -\frac {2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 125, 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 = {646, 43} \begin {gather*} -\frac {2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2}{2 b^3 (a+b x) \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
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\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)^2}{\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)^2}{b^7 (a+b x)^5}+\frac {2 e (b d-a e)}{b^7 (a+b x)^4}+\frac {e^2}{b^7 (a+b x)^3}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(b d-a e)^2}{4 b^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 e (b d-a e)}{3 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^2}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 69, normalized size = 0.55 \begin {gather*} \frac {-a^2 e^2-2 a b e (d+2 e x)-\left (b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )\right )}{12 b^3 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.48, size = 429, normalized size = 3.43 \begin {gather*} \frac {-2 \left (-3 a^6 b e^2+6 a^5 b^2 d e-3 a^4 b^3 d^2-a^2 b^5 e^2 x^4-2 a b^6 d e x^4-4 a b^6 e^2 x^5-3 b^7 d^2 x^4-8 b^7 d e x^5-6 b^7 e^2 x^6\right )-2 \sqrt {b^2} \sqrt {a^2+2 a b x+b^2 x^2} \left (-3 a^5 e^2+6 a^4 b d e+3 a^4 b e^2 x-3 a^3 b^2 d^2-6 a^3 b^2 d e x-3 a^3 b^2 e^2 x^2+3 a^2 b^3 d^2 x+6 a^2 b^3 d e x^2+3 a^2 b^3 e^2 x^3-3 a b^4 d^2 x^2-6 a b^4 d e x^3-2 a b^4 e^2 x^4+3 b^5 d^2 x^3+8 b^5 d e x^4+6 b^5 e^2 x^5\right )}{3 b^3 x^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-8 a^3 b^5-24 a^2 b^6 x-24 a b^7 x^2-8 b^8 x^3\right )+3 b^3 \sqrt {b^2} x^4 \left (8 a^4 b^4+32 a^3 b^5 x+48 a^2 b^6 x^2+32 a b^7 x^3+8 b^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 98, normalized size = 0.78 \begin {gather*} -\frac {6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 0.55 \begin {gather*} -\frac {\left (b x +a \right ) \left (6 b^{2} e^{2} x^{2}+4 a b \,e^{2} x +8 b^{2} d e x +a^{2} e^{2}+2 a b d e +3 b^{2} d^{2}\right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 115, normalized size = 0.92 \begin {gather*} -\frac {2 \, d e}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{2}} - \frac {e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, a e^{2}}{3 \, b^{6} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {d^{2}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {a d e}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{4}} - \frac {a^{2} e^{2}}{4 \, b^{7} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 79, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^2\,e^2+2\,a\,b\,d\,e+4\,a\,b\,e^2\,x+3\,b^2\,d^2+8\,b^2\,d\,e\,x+6\,b^2\,e^2\,x^2\right )}{12\,b^3\,{\left (a+b\,x\right )}^5} \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 )^{2}}{\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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