Optimal. Leaf size=172 \[ \frac {\left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{7/2}}-\frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (-4 a A c-8 a b B+5 A b^2\right )}{64 a^3 x^2}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4} \]
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Rubi [A] time = 0.14, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {834, 806, 720, 724, 206} \begin {gather*} -\frac {(2 a+b x) \sqrt {a+b x+c x^2} \left (-4 a A c-8 a b B+5 A b^2\right )}{64 a^3 x^2}+\frac {\left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{7/2}}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^5} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}-\frac {\int \frac {\left (\frac {1}{2} (5 A b-8 a B)+A c x\right ) \sqrt {a+b x+c x^2}}{x^4} \, dx}{4 a}\\ &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}+\frac {\left (5 A b^2-8 a b B-4 a A c\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{16 a^2}\\ &=-\frac {\left (5 A b^2-8 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{64 a^3 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}-\frac {\left (\left (b^2-4 a c\right ) \left (5 A b^2-8 a b B-4 a A c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^3}\\ &=-\frac {\left (5 A b^2-8 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{64 a^3 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}+\frac {\left (\left (b^2-4 a c\right ) \left (5 A b^2-8 a b B-4 a A c\right )\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 )}{64 a^3}\\ &=-\frac {\left (5 A b^2-8 a b B-4 a A c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{64 a^3 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4}+\frac {(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}+\frac {\left (b^2-4 a c\right ) \left (5 A b^2-8 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 153, normalized size = 0.89 \begin {gather*} \frac {\frac {3 \left (-4 a A c-8 a b B+5 A b^2\right ) \left (x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}\right )}{16 a^{3/2} x^2}+\frac {(5 A b-8 a B) (a+x (b+c x))^{3/2}}{x^3}-\frac {6 a A (a+x (b+c x))^{3/2}}{x^4}}{24 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.49, size = 229, normalized size = 1.33 \begin {gather*} -\frac {5 A b^4 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{64 a^{7/2}}+\frac {\left (2 a A c^2+4 a b B c-3 A b^2 c+b^3 (-B)\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (-48 a^3 A-64 a^3 B x-8 a^2 A b x-24 a^2 A c x^2-16 a^2 b B x^2-64 a^2 B c x^3+10 a A b^2 x^2+52 a A b c x^3+24 a b^2 B x^3-15 A b^3 x^3\right )}{192 a^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 425, normalized size = 2.47 \begin {gather*} \left [-\frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \, {\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} \sqrt {a} x^{4} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (48 \, A a^{4} - {\left (24 \, B a^{2} b^{2} - 15 \, A a b^{3} - 4 \, {\left (16 \, B a^{3} - 13 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2} + 12 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{4} x^{4}}, \frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \, {\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (48 \, A a^{4} - {\left (24 \, B a^{2} b^{2} - 15 \, A a b^{3} - 4 \, {\left (16 \, B a^{3} - 13 \, A a^{2} b\right )} c\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2} + 12 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{4} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 991, normalized size = 5.76 \begin {gather*} \frac {{\left (8 \, B a b^{3} - 5 \, A b^{4} - 32 \, B a^{2} b c + 24 \, A a b^{2} c - 16 \, A a^{2} c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{3}} - \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a^{2} b c + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a b^{2} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A a^{2} c^{2} - 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} c^{\frac {3}{2}} - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 55 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} - 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c - 264 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c - 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{3} c^{2} - 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{3} b^{2} \sqrt {c} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} - 1152 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{3} b c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} - 73 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c - 648 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c - 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{4} b^{2} \sqrt {c} - 384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} b^{3} \sqrt {c} - 128 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} - 256 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{4} b c^{\frac {3}{2}} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} b^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b^{4} + 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{5} b c - 312 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{2} c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} c^{2} + 128 \, B a^{6} c^{\frac {3}{2}} - 128 \, A a^{5} b c^{\frac {3}{2}}}{192 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{4} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 569, normalized size = 3.31 \begin {gather*} \frac {A \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {3}{2}}}-\frac {3 A \,b^{2} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {5}{2}}}+\frac {5 A \,b^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {7}{2}}}+\frac {B b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {3}{2}}}-\frac {B \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A b \,c^{2} x}{16 a^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} c x}{64 a^{4}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{2} c x}{8 a^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,c^{2}}{8 a^{2}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c}{32 a^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{64 a^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B b c}{4 a^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{8 a^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b c}{16 a^{3} x}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{64 a^{4} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{8 a^{3} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A c}{8 a^{2} x^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{32 a^{3} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{4 a^{2} x^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{24 a^{2} x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A}{4 a \,x^{4}} \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 = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^5} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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