Optimal. Leaf size=170 \[ \frac {3 \left (b^2-4 a c\right )^2 (A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{7/2}}-\frac {3 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {806, 720, 724, 206} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) (2 a+b x) (A b-2 a B) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {3 \left (b^2-4 a c\right )^2 (A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{7/2}}+\frac {(2 a+b x) (A b-2 a B) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}-\frac {(A b-2 a B) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx}{2 a}\\ &=\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}+\frac {\left (3 (A b-2 a B) \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{32 a^2}\\ &=-\frac {3 (A b-2 a B) \left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}-\frac {\left (3 (A b-2 a B) \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{256 a^3}\\ &=-\frac {3 (A b-2 a B) \left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}+\frac {\left (3 (A b-2 a B) \left (b^2-4 a c\right )^2\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 )}{128 a^3}\\ &=-\frac {3 (A b-2 a B) \left (b^2-4 a c\right ) (2 a+b x) \sqrt {a+b x+c x^2}}{128 a^3 x^2}+\frac {(A b-2 a B) (2 a+b x) \left (a+b x+c x^2\right )^{3/2}}{16 a^2 x^4}-\frac {A \left (a+b x+c x^2\right )^{5/2}}{5 a x^5}+\frac {3 (A b-2 a B) \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 157, normalized size = 0.92 \begin {gather*} \frac {(A b-2 a B) \left (16 a^{3/2} (2 a+b x) (a+x (b+c x))^{3/2}-3 x^2 \left (b^2-4 a c\right ) \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}-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 )\right )\right )}{256 a^{7/2} x^4}-\frac {A (a+x (b+c x))^{5/2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.51, size = 300, normalized size = 1.76 \begin {gather*} \frac {3 \left (8 a A b c^2+8 a b^2 B c-4 A b^3 c+b^4 (-B)\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{64 a^{5/2}}+\frac {3 \left (32 a^3 B c^2-A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{128 a^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-128 a^4 A-160 a^4 B x-176 a^3 A b x-256 a^3 A c x^2-240 a^3 b B x^2-400 a^3 B c x^3-8 a^2 A b^2 x^2-56 a^2 A b c x^3-128 a^2 A c^2 x^4-20 a^2 b^2 B x^3-200 a^2 b B c x^4+10 a A b^3 x^3+100 a A b^2 c x^4+30 a b^3 B x^4-15 A b^4 x^4\right )}{640 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.93, size = 555, normalized size = 3.26 \begin {gather*} \left [\frac {15 \, {\left (2 \, B a b^{4} - A b^{5} + 16 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} \sqrt {a} x^{5} \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 (128 \, A a^{5} - {\left (30 \, B a^{2} b^{3} - 15 \, A a b^{4} - 128 \, A a^{3} c^{2} - 100 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (10 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} + 4 \, {\left (50 \, B a^{4} + 7 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2} + 32 \, A a^{4} c\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2560 \, a^{4} x^{5}}, \frac {15 \, {\left (2 \, B a b^{4} - A b^{5} + 16 \, {\left (2 \, B a^{3} - A a^{2} b\right )} c^{2} - 8 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} c\right )} \sqrt {-a} x^{5} \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 (128 \, A a^{5} - {\left (30 \, B a^{2} b^{3} - 15 \, A a b^{4} - 128 \, A a^{3} c^{2} - 100 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (10 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3} + 4 \, {\left (50 \, B a^{4} + 7 \, A a^{3} b\right )} c\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2} + 32 \, A a^{4} c\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1280 \, a^{4} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 1357, normalized size = 7.98
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 978, normalized size = 5.75 \begin {gather*} \frac {3 A b \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}-\frac {3 A \,b^{3} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{32 a^{\frac {5}{2}}}+\frac {3 A \,b^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {7}{2}}}-\frac {3 B \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+\frac {3 B \,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 {3}{2}}}-\frac {3 B \,b^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {5}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c^{2} x}{32 a^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} c x}{128 a^{4}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b \,c^{2} x}{16 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} c x}{64 a^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b \,c^{2}}{16 a^{2}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} c}{64 a^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{128 a^{4}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} c^{2} x}{32 a^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{4} c x}{128 a^{5}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,c^{2}}{8 a}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} c}{32 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{64 a^{3}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b \,c^{2} x}{16 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} c x}{64 a^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b \,c^{2}}{16 a^{3}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3} c}{64 a^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{5}}{128 a^{5}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,c^{2}}{8 a^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} c}{32 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{64 a^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2} c}{32 a^{4} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{4}}{128 a^{5} x}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b c}{16 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{3}}{64 a^{4} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b c}{16 a^{3} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{3}}{64 a^{4} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{32 a^{3} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2}}{16 a^{3} x^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{8 a^{2} x^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{8 a^{2} x^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{4 a \,x^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{5 a \,x^{5}} \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 )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^6} \,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 ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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