Optimal. Leaf size=170 \[ -\frac {5 c^2 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c^2 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {686, 694, 329, 212, 206, 203} \[ -\frac {5 c^2 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c^2 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 329
Rule 686
Rule 694
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} \left (5 c d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (5 c^2 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx\\ &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{4} \left (5 c d^3\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 )\\ &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}+\frac {1}{2} \left (5 c d^3\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 )\\ &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {\left (5 c^2 d^4\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 )}{\sqrt {b^2-4 a c}}-\frac {\left (5 c^2 d^4\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 )}{\sqrt {b^2-4 a c}}\\ &=-\frac {d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 c d^3 \sqrt {b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac {5 c^2 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac {5 c^2 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )}{\left (b^2-4 a c\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 196, normalized size = 1.15 \[ \frac {(d (b+2 c x))^{7/2} \left (-64 \left (b^2-4 a c\right )^{3/4} (b+2 c x)^{5/2}+40 \left (b^2-4 a c\right )^{7/4} \sqrt {b+2 c x}+20 c (a+x (b+c x)) \left (2 \left (b^2-4 a c\right )^{3/4} \sqrt {b+2 c x}-12 c (a+x (b+c x)) \left (\tan ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )\right )\right )}{48 \left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2} (a+x (b+c x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 697, normalized size = 4.10 \[ \frac {20 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \arctan \left (-\frac {\left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {3}{4}} {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} \sqrt {2 \, c d x + b d} d^{3} - \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {3}{4}} \sqrt {2 \, c^{5} d^{7} x + b c^{4} d^{7} + \sqrt {\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}{c^{8} d^{14}}\right ) - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} + 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) + 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac {c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac {1}{4}} {\left (b^{2} - 4 \, a c\right )}\right ) - {\left (9 \, c^{2} d^{3} x^{2} + 9 \, b c d^{3} x + {\left (b^{2} + 5 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 510, normalized size = 3.00 \[ -\frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right )}{\sqrt {2} b^{2} - 4 \, \sqrt {2} a c} - \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {5 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c^{2} d^{3} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \, {\left (\sqrt {2} b^{2} - 4 \, \sqrt {2} a c\right )}} + \frac {2 \, {\left (5 \, \sqrt {2 \, c d x + b d} b^{2} c^{2} d^{7} - 20 \, \sqrt {2 \, c d x + b d} a c^{3} d^{7} - 9 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2} d^{5}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 435, normalized size = 2.56 \[ -\frac {40 \sqrt {2 c d x +b d}\, a \,c^{3} d^{7}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}+\frac {10 \sqrt {2 c d x +b d}\, b^{2} c^{2} d^{7}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}}-\frac {5 \sqrt {2}\, c^{2} d^{5} \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {5 \sqrt {2}\, c^{2} d^{5} \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )}{2 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}+\frac {5 \sqrt {2}\, c^{2} d^{5} \ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )}{4 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}-\frac {18 \left (2 c d x +b d \right )^{\frac {5}{2}} c^{2} d^{5}}{\left (4 c^{2} d^{2} x^{2}+4 b c \,d^{2} x +4 a c \,d^{2}\right )^{2}} \]
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 = 0.59, size = 309, normalized size = 1.82 \[ -\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (40\,a\,c^3\,d^7-10\,b^2\,c^2\,d^7\right )+18\,c^2\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4-{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (2\,b^2\,d^2-8\,a\,c\,d^2\right )+b^4\,d^4+16\,a^2\,c^2\,d^4-8\,a\,b^2\,c\,d^4}-\frac {5\,c^2\,d^{7/2}\,\mathrm {atan}\left (\frac {2000\,c^6\,d^{27/2}\,\sqrt {b\,d+2\,c\,d\,x}}{\left (\frac {2000\,b^2\,c^6\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}-\frac {8000\,a\,c^7\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{3/4}}-\frac {5\,c^2\,d^{7/2}\,\mathrm {atanh}\left (\frac {2000\,c^6\,d^{27/2}\,\sqrt {b\,d+2\,c\,d\,x}}{\left (\frac {2000\,b^2\,c^6\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}-\frac {8000\,a\,c^7\,d^{14}}{{\left (b^2-4\,a\,c\right )}^{3/2}}\right )\,{\left (b^2-4\,a\,c\right )}^{3/4}}\right )}{{\left (b^2-4\,a\,c\right )}^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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