Optimal. Leaf size=118 \[ \frac {16 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac {8 c}{d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {687, 688, 205} \[ \frac {16 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac {8 c}{d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 687
Rule 688
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac {(4 c) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (16 c^2\right ) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (64 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {16 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 62, normalized size = 0.53 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{3 d \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.99, size = 620, normalized size = 5.25 \[ \left [\frac {2 \, {\left (12 \, {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + {\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}, \frac {2 \, {\left (24 \, {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 906, normalized size = 7.68 \[ \frac {32 \, c^{2} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} \sqrt {b^{2} c - 4 \, a c^{2}}} + \frac {2 \, {\left (12 \, {\left (\frac {{\left (b^{16} c^{2} d^{3} - 32 \, a b^{14} c^{3} d^{3} + 448 \, a^{2} b^{12} c^{4} d^{3} - 3584 \, a^{3} b^{10} c^{5} d^{3} + 17920 \, a^{4} b^{8} c^{6} d^{3} - 57344 \, a^{5} b^{6} c^{7} d^{3} + 114688 \, a^{6} b^{4} c^{8} d^{3} - 131072 \, a^{7} b^{2} c^{9} d^{3} + 65536 \, a^{8} c^{10} d^{3}\right )} x}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}} + \frac {b^{17} c d^{3} - 32 \, a b^{15} c^{2} d^{3} + 448 \, a^{2} b^{13} c^{3} d^{3} - 3584 \, a^{3} b^{11} c^{4} d^{3} + 17920 \, a^{4} b^{9} c^{5} d^{3} - 57344 \, a^{5} b^{7} c^{6} d^{3} + 114688 \, a^{6} b^{5} c^{7} d^{3} - 131072 \, a^{7} b^{3} c^{8} d^{3} + 65536 \, a^{8} b c^{9} d^{3}}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}}\right )} x - \frac {b^{18} d^{3} - 48 \, a b^{16} c d^{3} + 960 \, a^{2} b^{14} c^{2} d^{3} - 10752 \, a^{3} b^{12} c^{3} d^{3} + 75264 \, a^{4} b^{10} c^{4} d^{3} - 344064 \, a^{5} b^{8} c^{5} d^{3} + 1032192 \, a^{6} b^{6} c^{6} d^{3} - 1966080 \, a^{7} b^{4} c^{7} d^{3} + 2162688 \, a^{8} b^{2} c^{8} d^{3} - 1048576 \, a^{9} c^{9} d^{3}}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 207, normalized size = 1.75 \[ -\frac {16 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, d}+\frac {8 c}{\left (4 a c -b^{2}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, d}+\frac {2}{3 \left (4 a c -b^{2}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} d} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (b\,d+2\,c\,d\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{2} b \sqrt {a + b x + c x^{2}} + 2 a^{2} c x \sqrt {a + b x + c x^{2}} + 2 a b^{2} x \sqrt {a + b x + c x^{2}} + 6 a b c x^{2} \sqrt {a + b x + c x^{2}} + 4 a c^{2} x^{3} \sqrt {a + b x + c x^{2}} + b^{3} x^{2} \sqrt {a + b x + c x^{2}} + 4 b^{2} c x^{3} \sqrt {a + b x + c x^{2}} + 5 b c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 2 c^{3} x^{5} \sqrt {a + b x + c x^{2}}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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