Optimal. Leaf size=67 \[ -\frac {4 b \sqrt {c+d x} (b c-a d)}{d^3}-\frac {2 (b c-a d)^2}{d^3 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{3/2}}{3 d^3} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ -\frac {4 b \sqrt {c+d x} (b c-a d)}{d^3}-\frac {2 (b c-a d)^2}{d^3 \sqrt {c+d x}}+\frac {2 b^2 (c+d x)^{3/2}}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{(c+d x)^{3/2}} \, dx &=\int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^{3/2}}-\frac {2 b (b c-a d)}{d^2 \sqrt {c+d x}}+\frac {b^2 \sqrt {c+d x}}{d^2}\right ) \, dx\\ &=-\frac {2 (b c-a d)^2}{d^3 \sqrt {c+d x}}-\frac {4 b (b c-a d) \sqrt {c+d x}}{d^3}+\frac {2 b^2 (c+d x)^{3/2}}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.88 \[ \frac {2 \left (-3 a^2 d^2+6 a b d (2 c+d x)+b^2 \left (-8 c^2-4 c d x+d^2 x^2\right )\right )}{3 d^3 \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 73, normalized size = 1.09 \[ \frac {2 \, {\left (b^{2} d^{2} x^{2} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x + c d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 84, normalized size = 1.25 \[ -\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt {d x + c} d^{3}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} b^{2} d^{6} - 6 \, \sqrt {d x + c} b^{2} c d^{6} + 6 \, \sqrt {d x + c} a b d^{7}\right )}}{3 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 63, normalized size = 0.94 \[ -\frac {2 \left (-b^{2} x^{2} d^{2}-6 a b \,d^{2} x +4 b^{2} c d x +3 a^{2} d^{2}-12 a b c d +8 b^{2} c^{2}\right )}{3 \sqrt {d x +c}\, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 75, normalized size = 1.12 \[ \frac {2 \, {\left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} b^{2} - 6 \, {\left (b^{2} c - a b d\right )} \sqrt {d x + c}}{d^{2}} - \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt {d x + c} d^{2}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 67, normalized size = 1.00 \[ \frac {\frac {2\,b^2\,{\left (c+d\,x\right )}^2}{3}-2\,a^2\,d^2-2\,b^2\,c^2-4\,b^2\,c\,\left (c+d\,x\right )+4\,a\,b\,d\,\left (c+d\,x\right )+4\,a\,b\,c\,d}{d^3\,\sqrt {c+d\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.29, size = 65, normalized size = 0.97 \[ \frac {2 b^{2} \left (c + d x\right )^{\frac {3}{2}}}{3 d^{3}} + \frac {\sqrt {c + d x} \left (4 a b d - 4 b^{2} c\right )}{d^{3}} - \frac {2 \left (a d - b c\right )^{2}}{d^{3} \sqrt {c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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