Optimal. Leaf size=273 \[ -\frac {5 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2}}-\frac {5}{8} \sqrt {a} \left (4 a A c+4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {5 \sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{64 c}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{24 x} \]
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Rubi [A] time = 0.26, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {812, 814, 843, 621, 206, 724} \begin {gather*} -\frac {5 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2}}+\frac {5 \sqrt {a+b x+c x^2} \left (2 c x \left (12 a B c+16 A b c+b^2 B\right )+32 a A c^2+44 a b B c+40 A b^2 c+b^3 B\right )}{64 c}-\frac {5}{8} \sqrt {a} \left (4 a A c+4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2} (6 (a B+A b)-x (4 A c+b B))}{24 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^3} \, dx &=-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5}{16} \int \frac {(-4 (A b+a B)-2 (b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}+\frac {5}{32} \int \frac {\left (4 \left (3 A b^2+4 a b B+4 a A c\right )+2 \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{x} \, dx\\ &=\frac {5 \left (b^3 B+40 A b^2 c+44 a b B c+32 a A c^2+2 c \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c}-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5 \int \frac {-16 a c \left (3 A b^2+4 a b B+4 a A c\right )+\left (b^4 B-8 A b^3 c-24 a b^2 B c-96 a A b c^2-48 a^2 B c^2\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{128 c}\\ &=\frac {5 \left (b^3 B+40 A b^2 c+44 a b B c+32 a A c^2+2 c \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c}-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}+\frac {1}{8} \left (5 a \left (3 A b^2+4 a b B+4 a A c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx-\frac {\left (5 \left (b^4 B-8 A b^3 c-24 a b^2 B c-96 a A b c^2-48 a^2 B c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c}\\ &=\frac {5 \left (b^3 B+40 A b^2 c+44 a b B c+32 a A c^2+2 c \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c}-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {1}{4} \left (5 a \left (3 A b^2+4 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 )-\frac {\left (5 \left (b^4 B-8 A b^3 c-24 a b^2 B c-96 a A b c^2-48 a^2 B 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 )}{64 c}\\ &=\frac {5 \left (b^3 B+40 A b^2 c+44 a b B c+32 a A c^2+2 c \left (b^2 B+16 A b c+12 a B c\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c}-\frac {5 (6 (A b+a B)-(b B+4 A c) x) \left (a+b x+c x^2\right )^{3/2}}{24 x}-\frac {(2 A-B x) \left (a+b x+c x^2\right )^{5/2}}{4 x^2}-\frac {5}{8} \sqrt {a} \left (3 A b^2+4 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 )-\frac {5 \left (b^4 B-8 A b^3 c-24 a b^2 B c-96 a A b c^2-48 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 254, normalized size = 0.93 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (-96 a^2 c (A+2 B x)+4 a c x (B x (139 b+54 c x)-4 A (27 b-28 c x))+x^2 \left (2 b^2 c (132 A+59 B x)+8 b c^2 x (26 A+17 B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )\right )}{192 c x^2}+\frac {5 \left (48 a^2 B c^2+96 a A b c^2+24 a b^2 B c+8 A b^3 c+b^4 (-B)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{128 c^{3/2}}-\frac {5}{8} \sqrt {a} \left (4 a A c+4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.12, size = 287, normalized size = 1.05 \begin {gather*} -\frac {5}{4} \left (4 a^{3/2} A c+4 a^{3/2} b B+3 \sqrt {a} A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )+\frac {\sqrt {a+b x+c x^2} \left (-96 a^2 A c-192 a^2 B c x-432 a A b c x+448 a A c^2 x^2+556 a b B c x^2+216 a B c^2 x^3+264 A b^2 c x^2+208 A b c^2 x^3+64 A c^3 x^4+15 b^3 B x^2+118 b^2 B c x^3+136 b B c^2 x^4+48 B c^3 x^5\right )}{192 c x^2}+\frac {5 \left (-48 a^2 B c^2-96 a A b c^2-24 a b^2 B c-8 A b^3 c+b^4 B\right ) \log \left (-2 c^{3/2} \sqrt {a+b x+c x^2}+b c+2 c^2 x\right )}{128 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 5.68, size = 1269, normalized size = 4.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 527, normalized size = 1.93 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, B c^{2} x + \frac {17 \, B b c^{4} + 8 \, A c^{5}}{c^{3}}\right )} x + \frac {59 \, B b^{2} c^{3} + 108 \, B a c^{4} + 104 \, A b c^{4}}{c^{3}}\right )} x + \frac {15 \, B b^{3} c^{2} + 556 \, B a b c^{3} + 264 \, A b^{2} c^{3} + 448 \, A a c^{4}}{c^{3}}\right )} + \frac {5 \, {\left (4 \, B a^{2} b + 3 \, A a b^{2} + 4 \, A a^{2} c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} + \frac {5 \, {\left (B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c - 48 \, B a^{2} c^{2} - 96 \, A a b c^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {3}{2}}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} \sqrt {c} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} c - 8 \, B a^{4} \sqrt {c} - 16 \, A a^{3} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 663, normalized size = 2.43 \begin {gather*} -\frac {5 A \,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 A 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 A \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 A \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}+\frac {15 B \,a^{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}} b \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^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {5 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {3}{2}}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A b c x}{2}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B a c x}{8}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x}{32}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A a c}{2}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b c x}{4 a}+\frac {25 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{8}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, B a b}{16}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B c x}{4}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{4 a}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b c x}{4 a^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A c}{6}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B c x}{a}+\frac {35 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{24}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A c}{2 a}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2}}{4 a^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{a}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A b}{4 a^{2} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} B}{a x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A}{2 a \,x^{2}} \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^3} \,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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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