Optimal. Leaf size=350 \[ a^{5/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\sqrt {a+b x+c x^2} \left (512 a^2 A c^3+2 c x \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )+b \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )\right )}{512 c^3}+\frac {\left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (80 a^2 B c^2+112 a A b c^2-40 a b^2 B c-12 A b^3 c+5 b^4 B\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{7/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 c x \left (-20 a B c-12 A b c+5 b^2 B\right )-64 a A c^2-20 a b B c-12 A b^2 c+5 b^3 B\right )}{192 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (12 A c+5 b B+10 B c x)}{60 c} \]
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Rubi [A] time = 0.42, antiderivative size = 350, 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 = {814, 843, 621, 206, 724} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (512 a^2 A c^3+2 c x \left (\left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b c^2\right )+b \left (b^2-4 a c\right ) \left (-20 a B c-12 A b c+5 b^2 B\right )+64 a A b^2 c^2\right )}{512 c^3}+\frac {\left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (80 a^2 B c^2+112 a A b c^2-40 a b^2 B c-12 A b^3 c+5 b^4 B\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{7/2}}+a^{5/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} \left (2 c x \left (-20 a B c-12 A b c+5 b^2 B\right )-64 a A c^2-20 a b B c-12 A b^2 c+5 b^3 B\right )}{192 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} (12 A c+5 b B+10 B c x)}{60 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x} \, dx &=\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}-\frac {\int \frac {\left (-12 a A c-\frac {1}{2} \left (12 A b c-5 B \left (b^2-4 a c\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx}{12 c}\\ &=-\frac {\left (5 b^3 B-12 A b^2 c-20 a b B c-64 a A c^2+2 c \left (5 b^2 B-12 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}+\frac {\int \frac {\left (96 a^2 A c^2+\frac {3}{4} \left (64 a A b c^2-\left (b^2-4 a c\right ) \left (12 A b c-5 B \left (b^2-4 a c\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{x} \, dx}{96 c^2}\\ &=\frac {\left (64 a A b^2 c^2+512 a^2 A c^3+b \left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )+2 c \left (64 a A b c^2+\left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 c^3}-\frac {\left (5 b^3 B-12 A b^2 c-20 a b B c-64 a A c^2+2 c \left (5 b^2 B-12 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}-\frac {\int \frac {-384 a^3 A c^3-\frac {3}{8} \left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (5 b^4 B-12 A b^3 c-40 a b^2 B c+112 a A b c^2+80 a^2 B c^2\right )\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{384 c^3}\\ &=\frac {\left (64 a A b^2 c^2+512 a^2 A c^3+b \left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )+2 c \left (64 a A b c^2+\left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 c^3}-\frac {\left (5 b^3 B-12 A b^2 c-20 a b B c-64 a A c^2+2 c \left (5 b^2 B-12 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}+\left (a^3 A\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx+\frac {\left (320 a^3 B-\frac {b^5 (5 b B-12 A c)}{c^3}+\frac {20 a b^3 (3 b B-8 A c)}{c^2}-\frac {240 a^2 b (b B-4 A c)}{c}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024}\\ &=\frac {\left (64 a A b^2 c^2+512 a^2 A c^3+b \left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )+2 c \left (64 a A b c^2+\left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 c^3}-\frac {\left (5 b^3 B-12 A b^2 c-20 a b B c-64 a A c^2+2 c \left (5 b^2 B-12 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}-\left (2 a^3 A\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 {1}{512} \left (320 a^3 B-\frac {b^5 (5 b B-12 A c)}{c^3}+\frac {20 a b^3 (3 b B-8 A c)}{c^2}-\frac {240 a^2 b (b B-4 A c)}{c}\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 (64 a A b^2 c^2+512 a^2 A c^3+b \left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )+2 c \left (64 a A b c^2+\left (b^2-4 a c\right ) \left (5 b^2 B-12 A b c-20 a B c\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{512 c^3}-\frac {\left (5 b^3 B-12 A b^2 c-20 a b B c-64 a A c^2+2 c \left (5 b^2 B-12 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^2}+\frac {(5 b B+12 A c+10 B c x) \left (a+b x+c x^2\right )^{5/2}}{60 c}-a^{5/2} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\left (320 a^3 B-\frac {b^5 (5 b B-12 A c)}{c^3}+\frac {20 a b^3 (3 b B-8 A c)}{c^2}-\frac {240 a^2 b (b B-4 A c)}{c}\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 349, normalized size = 1.00 \begin {gather*} a^{5/2} (-A) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )+\frac {\left (512 a^2 A b c^3-\left (b^2-4 a c\right ) \left (80 a^2 B c^2+112 a A b c^2-40 a b^2 B c-12 A b^3 c+5 b^4 B\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{1024 c^{7/2}}+\frac {\sqrt {a+x (b+c x)} \left (32 a^2 c^3 (16 A+5 B x)-8 b^3 c (5 a B+3 A c x)+16 a b^2 c^2 (7 A-5 B x)+16 a b c^2 (5 a B+14 A c x)-2 b^4 c (6 A-5 B x)+5 b^5 B\right )}{512 c^3}+\frac {(a+x (b+c x))^{3/2} \left (4 b c (5 a B+6 A c x)+8 a c^2 (8 A+5 B x)+2 b^2 c (6 A-5 B x)-5 b^3 B\right )}{192 c^2}+\frac {(a+x (b+c x))^{5/2} (2 c (6 A+5 B x)+5 b B)}{60 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.94, size = 369, normalized size = 1.05 \begin {gather*} 2 a^{5/2} A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+b x+c x^2} \left (11776 a^2 A c^3+2640 a^2 b B c^2+5280 a^2 B c^3 x+2160 a A b^2 c^2+9952 a A b c^3 x+5632 a A c^4 x^2-800 a b^3 B c+480 a b^2 B c^2 x+6240 a b B c^3 x^2+4160 a B c^4 x^3-180 A b^4 c+120 A b^3 c^2 x+2976 A b^2 c^3 x^2+4032 A b c^4 x^3+1536 A c^5 x^4+75 b^5 B-50 b^4 B c x+40 b^3 B c^2 x^2+2160 b^2 B c^3 x^3+3200 b B c^4 x^4+1280 B c^5 x^5\right )}{7680 c^3}+\frac {\left (-320 a^3 B c^3-960 a^2 A b c^3+240 a^2 b^2 B c^2+160 a A b^3 c^2-60 a b^4 B c-12 A b^5 c+5 b^6 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{1024 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 16.45, size = 1575, normalized size = 4.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 694, normalized size = 1.98 \begin {gather*} -A \,a^{\frac {5}{2}} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+\frac {15 A \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {5 A a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}+\frac {3 A \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {5}{2}}}+\frac {5 B \,a^{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} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {3}{2}}}+\frac {15 B a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {5}{2}}}-\frac {5 B \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {7}{2}}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a b x}{16}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} x}{64 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} x}{16}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2} x}{32 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4} x}{256 c^{2}}+\sqrt {c \,x^{2}+b x +a}\, A \,a^{2}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2}}{32 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{128 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b x}{8}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b}{32 c}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3}}{64 c^{2}}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a x}{24}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5}}{512 c^{3}}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} x}{96 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a}{3}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{16 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b}{48 c}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3}}{192 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B x}{6}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{5}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{12 c} \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} \,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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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