Optimal. Leaf size=67 \[ \frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {2 b^2 \sqrt {c+d x}}{d^3} \]
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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} \[ \frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {2 b^2 \sqrt {c+d x}}{d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{(c+d x)^{5/2}} \, dx &=\int \left (\frac {(-b c+a d)^2}{d^2 (c+d x)^{5/2}}-\frac {2 b (b c-a d)}{d^2 (c+d x)^{3/2}}+\frac {b^2}{d^2 \sqrt {c+d x}}\right ) \, dx\\ &=-\frac {2 (b c-a d)^2}{3 d^3 (c+d x)^{3/2}}+\frac {4 b (b c-a d)}{d^3 \sqrt {c+d x}}+\frac {2 b^2 \sqrt {c+d x}}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.93 \[ \frac {-2 a^2 d^2-4 a b d (2 c+3 d x)+2 b^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )}{3 d^3 (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 85, normalized size = 1.27 \[ \frac {2 \, {\left (3 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c^{2} - 4 \, a b c d - a^{2} d^{2} + 6 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x\right )} \sqrt {d x + c}}{3 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 72, normalized size = 1.07 \[ \frac {2 \, \sqrt {d x + c} b^{2}}{d^{3}} + \frac {2 \, {\left (6 \, {\left (d x + c\right )} b^{2} c - b^{2} c^{2} - 6 \, {\left (d x + c\right )} a b d + 2 \, a b c d - a^{2} d^{2}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 0.93 \[ -\frac {2 \left (-3 b^{2} x^{2} d^{2}+6 a b \,d^{2} x -12 b^{2} c d x +a^{2} d^{2}+4 a b c d -8 b^{2} c^{2}\right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 72, normalized size = 1.07 \[ \frac {2 \, {\left (\frac {3 \, \sqrt {d x + c} b^{2}}{d^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - 6 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 68, normalized size = 1.01 \[ \frac {6\,b^2\,{\left (c+d\,x\right )}^2-2\,a^2\,d^2-2\,b^2\,c^2+12\,b^2\,c\,\left (c+d\,x\right )-12\,a\,b\,d\,\left (c+d\,x\right )+4\,a\,b\,c\,d}{3\,d^3\,{\left (c+d\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 265, normalized size = 3.96 \[ \begin {cases} - \frac {2 a^{2} d^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} - \frac {8 a b c d}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} - \frac {12 a b d^{2} x}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {16 b^{2} c^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {24 b^{2} c d x}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} + \frac {6 b^{2} d^{2} x^{2}}{3 c d^{3} \sqrt {c + d x} + 3 d^{4} x \sqrt {c + d x}} & \text {for}\: d \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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