Optimal. Leaf size=88 \[ -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac {\left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}{16 c^3 d}+\frac {(b d+2 c d x)^{9/2}}{144 c^3 d^5} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac {\left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}{16 c^3 d}+\frac {(b d+2 c d x)^{9/2}}{144 c^3 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{\sqrt {b d+2 c d x}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 \sqrt {b d+2 c d x}}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{8 c^2 d^2}+\frac {(b d+2 c d x)^{7/2}}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}{16 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac {(b d+2 c d x)^{9/2}}{144 c^3 d^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 1.05 \begin {gather*} \frac {\left (c^2 \left (45 a^2+18 a c x^2+5 c^2 x^4\right )+3 b^2 c \left (c x^2-6 a\right )+2 b c^2 x \left (9 a+5 c x^2\right )+2 b^4-2 b^3 c x\right ) \sqrt {d (b+2 c x)}}{45 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 96, normalized size = 1.09 \begin {gather*} \frac {\left (45 a^2 c^2-18 a b^2 c+18 a b c^2 x+18 a c^3 x^2+2 b^4-2 b^3 c x+3 b^2 c^2 x^2+10 b c^3 x^3+5 c^4 x^4\right ) \sqrt {b d+2 c d x}}{45 c^3 d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 92, normalized size = 1.05 \begin {gather*} \frac {{\left (5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 18 \, a b^{2} c + 45 \, a^{2} c^{2} + 3 \, {\left (b^{2} c^{2} + 6 \, a c^{3}\right )} x^{2} - 2 \, {\left (b^{3} c - 9 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{45 \, c^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 349, normalized size = 3.97 \begin {gather*} \frac {5040 \, \sqrt {2 \, c d x + b d} a^{2} - \frac {1680 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a b}{c d} + \frac {84 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{2}}{c^{2} d^{2}} + \frac {168 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a}{c d^{2}} - \frac {36 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b}{c^{2} d^{3}} + \frac {315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c^{2} d^{4}}}{5040 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 96, normalized size = 1.09 \begin {gather*} \frac {\left (2 c x +b \right ) \left (5 c^{4} x^{4}+10 b \,c^{3} x^{3}+18 a \,c^{3} x^{2}+3 x^{2} b^{2} c^{2}+18 a b \,c^{2} x -2 x \,b^{3} c +45 a^{2} c^{2}-18 a \,b^{2} c +2 b^{4}\right )}{45 \sqrt {2 c d x +b d}\, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.47, size = 351, normalized size = 3.99 \begin {gather*} \frac {5040 \, \sqrt {2 \, c d x + b d} a^{2} - 168 \, a {\left (\frac {10 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b}{c d} - \frac {15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}}{c d^{2}}\right )} + \frac {84 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{2}}{c^{2} d^{2}} - \frac {36 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b}{c^{2} d^{3}} + \frac {315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c^{2} d^{4}}}{5040 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 99, normalized size = 1.12 \begin {gather*} \frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (5\,{\left (b\,d+2\,c\,d\,x\right )}^4+45\,b^4\,d^4-18\,b^2\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2+720\,a^2\,c^2\,d^4+72\,a\,c\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2-360\,a\,b^2\,c\,d^4\right )}{720\,c^3\,d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 61.62, size = 668, normalized size = 7.59 \begin {gather*} \begin {cases} \frac {- \frac {a^{2} b}{\sqrt {b d + 2 c d x}} - \frac {a^{2} \left (- \frac {b d}{\sqrt {b d + 2 c d x}} - \sqrt {b d + 2 c d x}\right )}{d} - \frac {a b^{2} \left (- \frac {b d}{\sqrt {b d + 2 c d x}} - \sqrt {b d + 2 c d x}\right )}{c d} - \frac {3 a b \left (\frac {b^{2} d^{2}}{\sqrt {b d + 2 c d x}} + 2 b d \sqrt {b d + 2 c d x} - \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3}\right )}{2 c d^{2}} - \frac {a \left (- \frac {b^{3} d^{3}}{\sqrt {b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt {b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac {3}{2}} - \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{2 c d^{3}} - \frac {b^{3} \left (\frac {b^{2} d^{2}}{\sqrt {b d + 2 c d x}} + 2 b d \sqrt {b d + 2 c d x} - \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3}\right )}{4 c^{2} d^{2}} - \frac {b^{2} \left (- \frac {b^{3} d^{3}}{\sqrt {b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt {b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac {3}{2}} - \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{2 c^{2} d^{3}} - \frac {5 b \left (\frac {b^{4} d^{4}}{\sqrt {b d + 2 c d x}} + 4 b^{3} d^{3} \sqrt {b d + 2 c d x} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} + \frac {4 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{16 c^{2} d^{4}} - \frac {- \frac {b^{5} d^{5}}{\sqrt {b d + 2 c d x}} - 5 b^{4} d^{4} \sqrt {b d + 2 c d x} + \frac {10 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}} + \frac {5 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} - \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}}{16 c^{2} d^{5}}}{c} & \text {for}\: c \neq 0 \\\frac {\begin {cases} a^{2} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{3}}{3 b} & \text {otherwise} \end {cases}}{\sqrt {b d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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