Optimal. Leaf size=121 \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]
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Rubi [A] time = 0.05, antiderivative size = 121, 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} \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{5/2}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{9/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{13/2}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 83, normalized size = 0.69 \[ \frac {\left (-315 \left (b^2-4 a c\right ) (b+2 c x)^4+495 \left (b^2-4 a c\right )^2 (b+2 c x)^2-385 \left (b^2-4 a c\right )^3+77 (b+2 c x)^6\right ) (d (b+2 c x))^{3/2}}{73920 c^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 205, normalized size = 1.69 \[ \frac {{\left (154 \, c^{7} x^{7} + 539 \, b c^{6} x^{6} - 2 \, b^{7} + 30 \, a b^{5} c - 165 \, a^{2} b^{3} c^{2} + 385 \, a^{3} b c^{3} + 21 \, {\left (31 \, b^{2} c^{5} + 30 \, a c^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{3} c^{4} + 45 \, a b c^{5}\right )} x^{4} + 5 \, {\left (b^{4} c^{3} + 216 \, a b^{2} c^{4} + 198 \, a^{2} c^{5}\right )} x^{3} - 3 \, {\left (b^{5} c^{2} - 15 \, a b^{3} c^{3} - 495 \, a^{2} b c^{4}\right )} x^{2} + {\left (2 \, b^{6} c - 30 \, a b^{4} c^{2} + 165 \, a^{2} b^{2} c^{3} + 770 \, a^{3} c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d}}{1155 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 1165, normalized size = 9.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 174, normalized size = 1.44 \[ \frac {\left (2 c x +b \right ) \left (77 c^{6} x^{6}+231 b \,c^{5} x^{5}+315 a \,c^{5} x^{4}+210 b^{2} c^{4} x^{4}+630 a b \,c^{4} x^{3}+35 b^{3} c^{3} x^{3}+495 a^{2} c^{4} x^{2}+225 a \,b^{2} c^{3} x^{2}-15 b^{4} c^{2} x^{2}+495 a^{2} b \,c^{3} x -90 a \,b^{3} c^{2} x +6 b^{5} c x +385 a^{3} c^{3}-165 a^{2} b^{2} c^{2}+30 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{1155 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 127, normalized size = 1.05 \[ -\frac {315 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 495 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} d^{4} + 385 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} d^{6} - 77 \, {\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{73920 \, c^{4} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 111, normalized size = 0.92 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{15/2}}{960\,c^4\,d^7}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{11/2}\,\left (4\,a\,c-b^2\right )}{704\,c^4\,d^5}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (4\,a\,c-b^2\right )}^3}{192\,c^4\,d}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (4\,a\,c-b^2\right )}^2}{448\,c^4\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.52, size = 151, normalized size = 1.25 \[ \frac {\frac {\left (b d + 2 c d x\right )^{\frac {3}{2}} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{192 c^{3}} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{448 c^{3} d^{2}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {11}{2}}}{704 c^{3} d^{4}} + \frac {\left (b d + 2 c d x\right )^{\frac {15}{2}}}{960 c^{3} d^{6}}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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