Optimal. Leaf size=175 \[ -2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+4 d^5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2} \]
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Rubi [A] time = 0.21, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {692, 694, 329, 212, 206, 203} \begin {gather*} 4 d^5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {4}{9} d (b d+2 c d x)^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 692
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx &=\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\\ &=\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{2 c}\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c}\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-\left (2 \left (b^2-4 a c\right )^{5/2} d^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )-\left (2 \left (b^2-4 a c\right )^{5/2} d^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 157, normalized size = 0.90 \begin {gather*} \frac {4 d (d (b+2 c x))^{9/2} \left (\frac {1}{9} (b+2 c x)^{9/2}-\left (4 a c-b^2\right ) \left (\frac {1}{5} (b+2 c x)^{5/2}-\frac {1}{2} \left (4 a c-b^2\right ) \left (2 \sqrt {b+2 c x}-\sqrt [4]{b^2-4 a c} \left (\tan ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )\right )\right )\right )}{(b+2 c x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.52, size = 311, normalized size = 1.78 \begin {gather*} \frac {4}{45} \sqrt {b d+2 c d x} \left (720 a^2 c^2 d^5-396 a b^2 c d^5-144 a b c^2 d^5 x-144 a c^3 d^5 x^2+59 b^4 d^5+76 b^3 c d^5 x+156 b^2 c^2 d^5 x^2+160 b c^3 d^5 x^3+80 c^4 d^5 x^4\right )+(1-i) d^{11/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac {-\frac {(1+i) c \sqrt {d} x}{\sqrt [4]{b^2-4 a c}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b \sqrt {d}}{\sqrt [4]{b^2-4 a c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {d} \sqrt [4]{b^2-4 a c}}{\sqrt {b d+2 c d x}}\right )-(1-i) d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b d+2 c d x}}{\sqrt {d} \left (\sqrt {b^2-4 a c}+i b+2 i c x\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1241, normalized size = 7.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 733, normalized size = 4.19 \begin {gather*} 4 \, \sqrt {2 \, c d x + b d} b^{4} d^{5} - 32 \, \sqrt {2 \, c d x + b d} a b^{2} c d^{5} + 64 \, \sqrt {2 \, c d x + b d} a^{2} c^{2} d^{5} + \frac {4}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{3} - \frac {16}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c d^{3} + \frac {4}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1287, normalized size = 7.35
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 240, normalized size = 1.37 \begin {gather*} \frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}+4\,d^5\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-2\,d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}+d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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