Optimal. Leaf size=119 \[ 2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {4}{3} d (b d+2 c d x)^{3/2} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {692, 694, 329, 298, 203, 206} \[ 2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{5/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {4}{3} d (b d+2 c d x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 692
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx &=\frac {4}{3} d (b d+2 c d x)^{3/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=\frac {4}{3} d (b d+2 c d x)^{3/2}+\frac {\left (\left (b^2-4 a c\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c}\\ &=\frac {4}{3} d (b d+2 c d x)^{3/2}+\frac {\left (\left (b^2-4 a c\right ) d\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c}\\ &=\frac {4}{3} d (b d+2 c d x)^{3/2}-\left (2 \left (b^2-4 a c\right ) d^3\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 ) d^3\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 )\\ &=\frac {4}{3} d (b d+2 c d x)^{3/2}+2 \left (b^2-4 a c\right )^{3/4} d^{5/2} \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 )^{3/4} d^{5/2} \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 = 0.12, size = 104, normalized size = 0.87 \[ \frac {2 (d (b+2 c x))^{5/2} \left (3 \left (b^2-4 a c\right )^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\tanh ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )+2 (b+2 c x)^{3/2}\right )}{3 (b+2 c x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 602, normalized size = 5.06 \[ \frac {4}{3} \, {\left (2 \, c d^{2} x + b d^{2}\right )} \sqrt {2 \, c d x + b d} + 4 \, \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {1}{4}} \arctan \left (\frac {\left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {1}{4}} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{7} - \sqrt {2 \, {\left (b^{8} c - 16 \, a b^{6} c^{2} + 96 \, a^{2} b^{4} c^{3} - 256 \, a^{3} b^{2} c^{4} + 256 \, a^{4} c^{5}\right )} d^{15} x + {\left (b^{9} - 16 \, a b^{7} c + 96 \, a^{2} b^{5} c^{2} - 256 \, a^{3} b^{3} c^{3} + 256 \, a^{4} b c^{4}\right )} d^{15} + \sqrt {{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}} \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {1}{4}}}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}}\right ) - \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{7} + \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {3}{4}}\right ) + \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {1}{4}} \log \left ({\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{7} - \left ({\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{10}\right )^{\frac {3}{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 354, normalized size = 2.97 \[ -\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} d \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} d \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {4}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 582, normalized size = 4.89 \[ \frac {4 \sqrt {2}\, a c \,d^{3} \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {4 \sqrt {2}\, a c \,d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {2 \sqrt {2}\, a c \,d^{3} \ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, b^{2} d^{3} \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, b^{2} d^{3} \ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )}{2 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+\frac {4 \left (2 c d x +b d \right )^{\frac {3}{2}} d}{3} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 97, normalized size = 0.82 \[ \frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{3}+2\,d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}-2\,d^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 97.72, size = 700, normalized size = 5.88 \[ 32 a b c d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left (t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 32 a b c d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left (t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 16 a c d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left (t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 8 b^{3} d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left (t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 8 b^{3} d^{4} \operatorname {RootSum} {\left (t^{4} \left (16384 a^{3} c^{3} d^{6} - 12288 a^{2} b^{2} c^{2} d^{6} + 3072 a b^{4} c d^{6} - 256 b^{6} d^{6}\right ) + 1, \left (t \mapsto t \log {\left (16 t a c d^{2} - 4 t b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 8 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left (t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} - 8 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left (t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + 4 b^{2} d^{3} \operatorname {RootSum} {\left (t^{4} \left (1024 a c d^{2} - 256 b^{2} d^{2}\right ) + 1, \left (t \mapsto t \log {\left (256 t^{3} a c d^{2} - 64 t^{3} b^{2} d^{2} + \sqrt {b d + 2 c d x} \right )} \right )\right )} + \frac {4 d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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