Optimal. Leaf size=153 \[ \frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2}-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)} \]
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Rubi [A] time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {684, 612, 621, 206} \begin {gather*} -\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 684
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^2} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {5 \int \left (a+b x+c x^2\right )^{3/2} \, dx}{4 c d^2}\\ &=\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}-\frac {\left (15 \left (b^2-4 a c\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{64 c^2 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{512 c^3 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {\left (15 \left (b^2-4 a c\right )^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 )}{256 c^3 d^2}\\ &=-\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 97, normalized size = 0.63 \begin {gather*} -\frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 d^2 (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.83, size = 165, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-128 a^2 c^2+100 a b^2 c+144 a b c^2 x+144 a c^3 x^2-15 b^4-20 b^3 c x+12 b^2 c^2 x^2+64 b c^3 x^3+32 c^4 x^4\right )}{256 c^3 d^2 (b+2 c x)}-\frac {15 \left (16 a^2 c^2-8 a b^2 c+b^4\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{512 c^{7/2} d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 427, normalized size = 2.79 \begin {gather*} \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (32 \, c^{5} x^{4} + 64 \, b c^{4} x^{3} - 15 \, b^{4} c + 100 \, a b^{2} c^{2} - 128 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{2} - 4 \, {\left (5 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1024 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (32 \, c^{5} x^{4} + 64 \, b c^{4} x^{3} - 15 \, b^{4} c + 100 \, a b^{2} c^{2} - 128 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 12 \, a c^{4}\right )} x^{2} - 4 \, {\left (5 \, b^{3} c^{2} - 36 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{512 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 814, normalized size = 5.32
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 961, normalized size = 6.28 \begin {gather*} \frac {15 a^{3} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 \left (4 a c -b^{2}\right ) \sqrt {c}\, d^{2}}-\frac {45 a^{2} b^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{32 \left (4 a c -b^{2}\right ) c^{\frac {3}{2}} d^{2}}+\frac {45 a \,b^{4} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{128 \left (4 a c -b^{2}\right ) c^{\frac {5}{2}} d^{2}}-\frac {15 b^{6} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{512 \left (4 a c -b^{2}\right ) c^{\frac {7}{2}} d^{2}}+\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} x}{8 \left (4 a c -b^{2}\right ) d^{2}}-\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{2} x}{16 \left (4 a c -b^{2}\right ) c \,d^{2}}+\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{4} x}{128 \left (4 a c -b^{2}\right ) c^{2} d^{2}}+\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} b}{16 \left (4 a c -b^{2}\right ) c \,d^{2}}-\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{3}}{32 \left (4 a c -b^{2}\right ) c^{2} d^{2}}+\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a x}{4 \left (4 a c -b^{2}\right ) d^{2}}+\frac {15 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{5}}{256 \left (4 a c -b^{2}\right ) c^{3} d^{2}}-\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2} x}{16 \left (4 a c -b^{2}\right ) c \,d^{2}}+\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a b}{8 \left (4 a c -b^{2}\right ) c \,d^{2}}-\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{3}}{32 \left (4 a c -b^{2}\right ) c^{2} d^{2}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}} x}{\left (4 a c -b^{2}\right ) d^{2}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}} b}{2 \left (4 a c -b^{2}\right ) c \,d^{2}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right ) c \,d^{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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^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*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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