Optimal. Leaf size=255 \[ -\frac {5 \sqrt {a+b x+c x^2} \left (-x \left (4 a B c+4 A b c+b^2 B\right )+A \left (4 a c+b^2\right )+4 a b B\right )}{8 x}+\frac {5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c}}-\frac {5 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {a}}-\frac {5 \left (a+b x+c x^2\right )^{3/2} (2 a B-x (2 A c+b B)+A b)}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3} \]
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Rubi [A] time = 0.32, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {812, 843, 621, 206, 724} \begin {gather*} \frac {5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c}}-\frac {5 \sqrt {a+b x+c x^2} \left (-x \left (4 a B c+4 A b c+b^2 B\right )+A \left (4 a c+b^2\right )+4 a b B\right )}{8 x}-\frac {5 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {a}}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}-\frac {5 \left (a+b x+c x^2\right )^{3/2} (2 a B-x (2 A c+b B)+A b)}{12 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^4} \, dx &=-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}-\frac {5}{18} \int \frac {(-3 (A b+2 a B)-3 (b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac {5 (A b+2 a B-(b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}+\frac {5}{48} \int \frac {\left (6 \left (4 a b B+A \left (b^2+4 a c\right )\right )+6 \left (b^2 B+4 A b c+4 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{x^2} \, dx\\ &=-\frac {5 \left (4 a b B+A \left (b^2+4 a c\right )-\left (b^2 B+4 A b c+4 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{8 x}-\frac {5 (A b+2 a B-(b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}-\frac {5}{96} \int \frac {-6 \left (2 a B \left (3 b^2+4 a c\right )+A \left (b^3+12 a b c\right )\right )-6 \left (b^3 B+6 A b^2 c+12 a b B c+8 a A c^2\right ) x}{x \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {5 \left (4 a b B+A \left (b^2+4 a c\right )-\left (b^2 B+4 A b c+4 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{8 x}-\frac {5 (A b+2 a B-(b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}+\frac {1}{16} \left (5 \left (b^3 B+6 A b^2 c+12 a b B c+8 a A c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx+\frac {1}{16} \left (5 \left (2 a B \left (3 b^2+4 a c\right )+A \left (b^3+12 a b c\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {5 \left (4 a b B+A \left (b^2+4 a c\right )-\left (b^2 B+4 A b c+4 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{8 x}-\frac {5 (A b+2 a B-(b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}+\frac {1}{8} \left (5 \left (b^3 B+6 A b^2 c+12 a b B c+8 a A c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )-\frac {1}{8} \left (5 \left (2 a B \left (3 b^2+4 a c\right )+A \left (b^3+12 a b c\right )\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 )\\ &=-\frac {5 \left (4 a b B+A \left (b^2+4 a c\right )-\left (b^2 B+4 A b c+4 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{8 x}-\frac {5 (A b+2 a B-(b B+2 A c) x) \left (a+b x+c x^2\right )^{3/2}}{12 x^2}-\frac {(A-B x) \left (a+b x+c x^2\right )^{5/2}}{3 x^3}-\frac {5 \left (2 a B \left (3 b^2+4 a c\right )+A \left (b^3+12 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 \sqrt {a}}+\frac {5 \left (b^3 B+6 A b^2 c+12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 236, normalized size = 0.93 \begin {gather*} \frac {1}{48} \left (\frac {2 \sqrt {a+x (b+c x)} \left (-4 a^2 (2 A+3 B x)-2 a x (A (13 b+28 c x)+B x (27 b-28 c x))+x^2 \left (A \left (-33 b^2+54 b c x+12 c^2 x^2\right )+B x \left (33 b^2+26 b c x+8 c^2 x^2\right )\right )\right )}{x^3}+\frac {15 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}-\frac {15 \left (A \left (12 a b c+b^3\right )+2 a B \left (4 a c+3 b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.53, size = 246, normalized size = 0.96 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-8 a^2 A-12 a^2 B x-26 a A b x-56 a A c x^2-54 a b B x^2+56 a B c x^3-33 A b^2 x^2+54 A b c x^3+12 A c^2 x^4+33 b^2 B x^3+26 b B c x^4+8 B c^2 x^5\right )}{24 x^3}-\frac {5 \left (8 a^2 B c+12 a A b c+6 a b^2 B+A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {5 \left (8 a A c^2+12 a b B c+6 A b^2 c+b^3 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{16 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.27, size = 1293, normalized size = 5.07
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 784, normalized size = 3.07 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, B c^{2} x + \frac {13 \, B b c^{3} + 6 \, A c^{4}}{c^{2}}\right )} x + \frac {33 \, B b^{2} c^{2} + 56 \, B a c^{3} + 54 \, A b c^{3}}{c^{2}}\right )} + \frac {5 \, {\left (6 \, B a b^{2} + A b^{3} + 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} - \frac {5 \, {\left (B b^{3} + 12 \, B a b c + 6 \, A b^{2} c + 8 \, A a c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, \sqrt {c}} + \frac {54 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} \sqrt {c} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} \sqrt {c} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c^{\frac {3}{2}} + 108 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c^{\frac {3}{2}} + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{2} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} \sqrt {c} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} \sqrt {c} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b c^{\frac {3}{2}} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b c - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b^{2} c - 192 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{2} + 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} \sqrt {c} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} \sqrt {c} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c^{\frac {3}{2}} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c^{\frac {3}{2}} + 96 \, B a^{4} b c + 48 \, A a^{3} b^{2} c + 112 \, A a^{4} c^{2}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 840, normalized size = 3.29 \begin {gather*} \frac {5 A a \,c^{\frac {3}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}-\frac {15 A \sqrt {a}\, b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4}-\frac {5 A \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 \sqrt {a}}+\frac {15 A \,b^{2} \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8}-\frac {5 B \,a^{\frac {3}{2}} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2}+\frac {15 B a b \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4}-\frac {15 B \sqrt {a}\, b^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8}+\frac {5 B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c x}{8 a}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,c^{2} x}{2}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b c x}{2}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{8 a}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,c^{2} x}{3 a}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} c x}{24 a^{2}}+5 \sqrt {c \,x^{2}+b x +a}\, A b c +\frac {5 \sqrt {c \,x^{2}+b x +a}\, B a c}{2}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b c x}{4 a}+\frac {25 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{8}+\frac {25 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b c}{12 a}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{24 a^{2}}+\frac {4 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,c^{2} x}{3 a^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2} c x}{8 a^{3}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{4 a}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b c x}{4 a^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B c}{6}+\frac {17 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b c}{12 a^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{3}}{8 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B c}{2 a}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{4 a^{2}}-\frac {4 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A c}{3 a^{2} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A \,b^{2}}{8 a^{3} x}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} B b}{4 a^{2} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A b}{12 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} B}{2 a \,x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A}{3 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 {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{x^4} \,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 {5}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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