Optimal. Leaf size=69 \[ \frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}} \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} \frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^2}{e^2 (d+e x)^{7/2}}-\frac {2 b (b d-a e)}{e^2 (d+e x)^{5/2}}+\frac {b^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}+\frac {4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac {2 b^2}{e^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.88 \begin {gather*} -\frac {2 \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 72, normalized size = 1.04 \begin {gather*} -\frac {2 \left (3 a^2 e^2+10 a b e (d+e x)-6 a b d e+3 b^2 d^2+15 b^2 (d+e x)^2-10 b^2 d (d+e x)\right )}{15 e^3 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 95, normalized size = 1.38 \begin {gather*} -\frac {2 \, {\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 4 \, a b d e + 3 \, a^{2} e^{2} + 10 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 72, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (x e + d\right )}^{2} b^{2} - 10 \, {\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} + 10 \, {\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 63, normalized size = 0.91 \begin {gather*} -\frac {2 \left (15 b^{2} e^{2} x^{2}+10 a b \,e^{2} x +20 b^{2} d e x +3 a^{2} e^{2}+4 a b d e +8 b^{2} d^{2}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 65, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} b^{2} + 3 \, b^{2} d^{2} - 6 \, a b d e + 3 \, a^{2} e^{2} - 10 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 62, normalized size = 0.90 \begin {gather*} -\frac {6\,a^2\,e^2+8\,a\,b\,d\,e+20\,a\,b\,e^2\,x+16\,b^2\,d^2+40\,b^2\,d\,e\,x+30\,b^2\,e^2\,x^2}{15\,e^3\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 389, normalized size = 5.64 \begin {gather*} \begin {cases} - \frac {6 a^{2} e^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {8 a b d e}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {20 a b e^{2} x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {16 b^{2} d^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {40 b^{2} d e x}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} - \frac {30 b^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt {d + e x} + 30 d e^{4} x \sqrt {d + e x} + 15 e^{5} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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