Optimal. Leaf size=317 \[ -\frac {\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{9/2}}-\frac {\sqrt {a+b x+c x^2} \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.39, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {822, 834, 806, 724, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (256 a^2 c^2-460 a b^2 c+105 b^4\right )\right )}{24 a^4 x \left (b^2-4 a c\right )}-\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-6 a b B+7 A b^2\right )}{3 a^2 x^3 \left (b^2-4 a c\right )}-\frac {\sqrt {a+b x+c x^2} \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right )}{12 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (6 a B \left (5 b^2-4 a c\right )-A \left (35 b^3-60 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{9/2}}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 834
Rubi steps
\begin {align*} \int \frac {A+B x}{x^4 \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} \left (-7 A b^2+6 a b B+16 a A c\right )-3 (A b-2 a B) c x}{x^4 \sqrt {a+b x+c x^2}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {2 \int \frac {\frac {1}{4} \left (-35 A b^3+30 a b^2 B+116 a A b c-72 a^2 B c\right )-c \left (7 A b^2-6 a b B-16 a A c\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{3 a^2 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt {a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\int \frac {\frac {1}{8} \left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right )+\frac {1}{4} c \left (6 a B \left (5 b^2-12 a c\right )-A \left (35 b^3-116 a b c\right )\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{3 a^3 \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt {a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}-\frac {\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a^4}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt {a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}+\frac {\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{8 a^4}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x^3 \sqrt {a+b x+c x^2}}-\frac {\left (7 A b^2-6 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{3 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\left (35 A b^3-30 a b^2 B-116 a A b c+72 a^2 B c\right ) \sqrt {a+b x+c x^2}}{12 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (6 a b B \left (15 b^2-52 a c\right )-A \left (105 b^4-460 a b^2 c+256 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{24 a^4 \left (b^2-4 a c\right ) x}+\frac {\left (35 A b^3-30 a b^2 B-60 a A b c+24 a^2 B c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 294, normalized size = 0.93 \begin {gather*} \frac {\frac {2 \sqrt {a} \left (-16 a^4 c (2 A+3 B x)+4 a^3 \left (2 A \left (b^2+7 b c x+16 c^2 x^2\right )+3 B x \left (b^2+10 b c x-12 c^2 x^2\right )\right )+2 a^2 x \left (A \left (-7 b^3-86 b^2 c x+244 b c^2 x^2+128 c^3 x^3\right )+3 b B x \left (-5 b^2+62 b c x+52 c^2 x^2\right )\right )-5 a b^2 x^2 \left (A \left (-7 b^2+106 b c x+92 c^2 x^2\right )+18 b B x (b+c x)\right )+105 A b^4 x^3 (b+c x)\right )}{x^3 \sqrt {a+x (b+c x)}}-3 \left (b^2-4 a c\right ) \left (5 A \left (7 b^3-12 a b c\right )+6 a B \left (4 a c-5 b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{48 a^{9/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.11, size = 380, normalized size = 1.20 \begin {gather*} -\frac {15 \left (2 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {\left (-24 a^2 B c-35 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {-32 a^4 A c-48 a^4 B c x+8 a^3 A b^2+56 a^3 A b c x+128 a^3 A c^2 x^2+12 a^3 b^2 B x+120 a^3 b B c x^2-144 a^3 B c^2 x^3-14 a^2 A b^3 x-172 a^2 A b^2 c x^2+488 a^2 A b c^2 x^3+256 a^2 A c^3 x^4-30 a^2 b^3 B x^2+372 a^2 b^2 B c x^3+312 a^2 b B c^2 x^4+35 a A b^4 x^2-530 a A b^3 c x^3-460 a A b^2 c^2 x^4-90 a b^4 B x^3-90 a b^3 B c x^4+105 A b^5 x^3+105 A b^4 c x^4}{24 a^4 x^3 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.06, size = 1093, normalized size = 3.45
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 798, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (\frac {{\left (B a^{5} b^{3} c - A a^{4} b^{4} c - 3 \, B a^{6} b c^{2} + 4 \, A a^{5} b^{2} c^{2} - 2 \, A a^{6} c^{3}\right )} x}{a^{8} b^{2} - 4 \, a^{9} c} + \frac {B a^{5} b^{4} - A a^{4} b^{5} - 4 \, B a^{6} b^{2} c + 5 \, A a^{5} b^{3} c + 2 \, B a^{7} c^{2} - 5 \, A a^{6} b c^{2}}{a^{8} b^{2} - 4 \, a^{9} c}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (30 \, B a b^{2} - 35 \, A b^{3} - 24 \, B a^{2} c + 60 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4}} - \frac {42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} - 57 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt {c} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 136 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b^{2} \sqrt {c} - 192 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 54 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 87 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 96 \, B a^{4} b \sqrt {c} - 144 \, A a^{3} b^{2} \sqrt {c} + 80 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 708, normalized size = 2.23 \begin {gather*} \frac {32 A \,c^{3} x}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {115 A \,b^{2} c^{2} x}{6 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {35 A \,b^{4} c x}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}+\frac {13 B b \,c^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {15 B \,b^{3} c x}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {16 A b \,c^{2}}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {115 A \,b^{3} c}{12 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {35 A \,b^{5}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{4}}+\frac {13 B \,b^{2} c}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {15 B \,b^{4}}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {15 A b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {7}{2}}}+\frac {35 A \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {9}{2}}}+\frac {3 B c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {15 B \,b^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {7}{2}}}+\frac {15 A b c}{4 \sqrt {c \,x^{2}+b x +a}\, a^{3}}-\frac {35 A \,b^{3}}{16 \sqrt {c \,x^{2}+b x +a}\, a^{4}}-\frac {3 B c}{2 \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {15 B \,b^{2}}{8 \sqrt {c \,x^{2}+b x +a}\, a^{3}}+\frac {4 A c}{3 \sqrt {c \,x^{2}+b x +a}\, a^{2} x}-\frac {35 A \,b^{2}}{24 \sqrt {c \,x^{2}+b x +a}\, a^{3} x}+\frac {5 B b}{4 \sqrt {c \,x^{2}+b x +a}\, a^{2} x}+\frac {7 A b}{12 \sqrt {c \,x^{2}+b x +a}\, a^{2} x^{2}}-\frac {B}{2 \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}}-\frac {A}{3 \sqrt {c \,x^{2}+b x +a}\, a \,x^{3}} \end {gather*}
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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \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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