Optimal. Leaf size=310 \[ -\frac {1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{128 c^2}+\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x} \]
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Rubi [A] time = 0.31, antiderivative size = 310, 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 {\sqrt {a+b x+c x^2} \left (-128 a^2 B c^2+2 c x \left (-120 a A c^2-28 a b B c-10 A b^2 c+3 b^3 B\right )-440 a A b c^2-28 a b^2 B c-10 A b^3 c+3 b^4 B\right )}{128 c^2}+\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}-\frac {1}{2} a^{3/2} (2 a B+5 A b) \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 (16 a B c+6 c x (10 A c+b B)+70 A b c+3 b^2 B\right )}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 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^2} \, dx &=-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}-\frac {1}{2} \int \frac {(-5 A b-2 a B-(b B+10 A c) x) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx\\ &=\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}+\frac {\int \frac {\left (8 a (5 A b+2 a B) c-\frac {1}{2} \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{x} \, dx}{16 c}\\ &=-\frac {\left (3 b^4 B-10 A b^3 c-28 a b^2 B c-440 a A b c^2-128 a^2 B c^2+2 c \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2}+\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}-\frac {\int \frac {-32 a^2 (5 A b+2 a B) c^2-\frac {1}{4} \left (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{64 c^2}\\ &=-\frac {\left (3 b^4 B-10 A b^3 c-28 a b^2 B c-440 a A b c^2-128 a^2 B c^2+2 c \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2}+\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}+\frac {1}{2} \left (a^2 (5 A b+2 a B)\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx+\frac {\left (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2}\\ &=-\frac {\left (3 b^4 B-10 A b^3 c-28 a b^2 B c-440 a A b c^2-128 a^2 B c^2+2 c \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2}+\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}-\left (a^2 (5 A b+2 a B)\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 (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\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 )}{128 c^2}\\ &=-\frac {\left (3 b^4 B-10 A b^3 c-28 a b^2 B c-440 a A b c^2-128 a^2 B c^2+2 c \left (3 b^3 B-10 A b^2 c-28 a b B c-120 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2}+\frac {\left (3 b^2 B+70 A b c+16 a B c+6 c (b B+10 A c) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c}-\frac {(5 A-B x) \left (a+b x+c x^2\right )^{5/2}}{5 x}-\frac {1}{2} a^{3/2} (5 A b+2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )+\frac {\left (3 b^5 B-10 A b^4 c-40 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2+480 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 288, normalized size = 0.93 \begin {gather*} -\frac {1}{2} a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )+\frac {\sqrt {a+x (b+c x)} \left (-128 a^2 c^2 (15 A-23 B x)+4 a c x \left (2 b c (695 A+311 B x)+4 c^2 x (135 A+88 B x)+135 b^2 B\right )+x \left (30 b^3 c (5 A+B x)+4 b^2 c^2 x (295 A+186 B x)+16 b c^3 x^2 (85 A+63 B x)+96 c^4 x^3 (5 A+4 B x)-45 b^4 B\right )\right )}{1920 c^2 x}+\frac {\left (480 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-40 a b^3 B c-10 A b^4 c+3 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{256 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.86, size = 329, normalized size = 1.06 \begin {gather*} \left (5 a^{3/2} A b+2 a^{5/2} B\right ) \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 (-1920 a^2 A c^2+2944 a^2 B c^2 x+5560 a A b c^2 x+2160 a A c^3 x^2+540 a b^2 B c x+2488 a b B c^2 x^2+1408 a B c^3 x^3+150 A b^3 c x+1180 A b^2 c^2 x^2+1360 A b c^3 x^3+480 A c^4 x^4-45 b^4 B x+30 b^3 B c x^2+744 b^2 B c^2 x^3+1008 b B c^3 x^4+384 B c^4 x^5\right )}{1920 c^2 x}+\frac {\left (-480 a^2 A c^3-240 a^2 b B c^2-240 a A b^2 c^2+40 a b^3 B c+10 A b^4 c-3 b^5 B\right ) \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{256 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 8.46, size = 1393, normalized size = 4.49
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 = 615, normalized size = 1.98 \begin {gather*} \frac {15 A \,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 A \,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 A 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 A \,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}}}-B \,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 B \,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 B 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 B \,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 {15 \sqrt {c \,x^{2}+b x +a}\, A a c x}{8}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{32}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, B a b x}{16}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} x}{64 c}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, A a b}{16}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c}+\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A c x}{4}+\sqrt {c \,x^{2}+b x +a}\, B \,a^{2}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2}}{32 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{128 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b x}{8}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A c x}{a}+\frac {35 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{24}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a}{3}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{16 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{5}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} A}{a x} \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^2} \,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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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