Optimal. Leaf size=217 \[ \frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )\right )}{64 a^2 x^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {810, 843, 621, 206, 724} \[ -\frac {\sqrt {a+b x+c x^2} \left (x \left (8 a B \left (8 a c+b^2\right )-3 A \left (b^3-4 a b c\right )\right )+2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )\right )}{64 a^2 x^2}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} (x (8 a B+3 A b)+6 a A)}{24 a x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}-\frac {\int \frac {\left (\frac {1}{2} \left (-8 a b B+3 A \left (b^2-4 a c\right )\right )-8 a B c x\right ) \sqrt {a+b x+c x^2}}{x^3} \, dx}{8 a}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\int \frac {\frac {1}{4} \left (-8 a b B \left (b^2-12 a c\right )+3 A \left (b^2-4 a c\right )^2\right )+32 a^2 B c^2 x}{x \sqrt {a+b x+c x^2}} \, dx}{32 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (B c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx-\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{128 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\left (2 B c^2\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 {\left (8 a b B \left (b^2-12 a c\right )-3 A \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 )}{64 a^2}\\ &=-\frac {\left (2 a \left (8 a b B-3 A \left (b^2-4 a c\right )\right )+\left (8 a B \left (b^2+8 a c\right )-3 A \left (b^3-4 a b c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 a^2 x^2}-\frac {(6 a A+(3 A b+8 a B) x) \left (a+b x+c x^2\right )^{3/2}}{24 a x^4}+\frac {\left (8 a b B \left (b^2-12 a c\right )-3 A \left (b^2-4 a c\right )^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2}}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.63, size = 202, normalized size = 0.93 \[ -\frac {\left (3 A \left (b^2-4 a c\right )^2+8 a b B \left (12 a c-b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{128 a^{5/2}}-\frac {\sqrt {a+x (b+c x)} \left (16 a^3 (3 A+4 B x)+8 a^2 x (3 A (3 b+5 c x)+2 B x (7 b+16 c x))+6 a b x^2 (A (b+10 c x)+4 b B x)-9 A b^3 x^3\right )}{192 a^2 x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 6.36, size = 1083, normalized size = 4.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 1019, normalized size = 4.70 \[ -B c^{\frac {3}{2}} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right ) - \frac {{\left (8 \, B a b^{3} - 3 \, A b^{4} - 96 \, B a^{2} b c + 24 \, A a b^{2} c - 48 \, 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^{2}} + \frac {24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} B a b^{3} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} A b^{4} + 480 \, {\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 + 240 \, {\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^{2} b^{2} \sqrt {c} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} B a^{3} c^{\frac {3}{2}} + 768 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} A a^{2} b c^{\frac {3}{2}} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b^{4} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{3} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a^{2} b^{2} c + 144 \, {\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} A a^{2} b^{3} \sqrt {c} - 1536 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{4} c^{\frac {3}{2}} - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{3} b^{3} + 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b^{4} + 288 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{4} b c + 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{3} b^{2} c + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{4} c^{2} + 1280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{5} c^{\frac {3}{2}} + 768 \, {\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} - 9 \, {\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 + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{4} b^{2} c + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{5} c^{2} - 512 \, B a^{6} c^{\frac {3}{2}}}{192 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{4} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 838, normalized size = 3.86 \[ -\frac {3 A \,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 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 {3}{2}}}-\frac {3 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 {5}{2}}}-\frac {3 B b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 \sqrt {a}}+\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 {3}{2}}}+B \,c^{\frac {3}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b \,c^{2} x}{16 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} c x}{64 a^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,c^{2} x}{a}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{2} c x}{8 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,c^{2}}{8 a}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c}{32 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{64 a^{3}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b \,c^{2} x}{16 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3} c x}{64 a^{4}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b c}{4 a}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{8 a^{2}}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,c^{2} x}{3 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} c x}{24 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,c^{2}}{8 a^{2}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} c}{32 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{4}}{64 a^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b c}{12 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3}}{24 a^{3}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b c}{16 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{3}}{64 a^{4} x}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B c}{3 a^{2} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{24 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2}}{32 a^{3} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{12 a^{2} x^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{8 a^{2} x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{4 a \,x^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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