Optimal. Leaf size=197 \[ \frac {\sqrt {a x^2+b x+c} \left (256 a^5 x^5+640 a^4 b x^4+832 a^4 c x^3+432 a^3 b^2 x^3+1248 a^3 b c x^2+1056 a^3 c^2 x+8 a^2 b^3 x^2+96 a^2 b^2 c x+528 a^2 b c^2-10 a b^4 x-160 a b^3 c+15 b^5\right )}{1536 a^3}-\frac {5 \left (64 a^3 c^3-48 a^2 b^2 c^2+12 a b^4 c-b^6\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{1024 a^{7/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 149, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {612, 621, 206} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{1024 a^{7/2}}+\frac {5 \left (b^2-4 a c\right )^2 (2 a x+b) \sqrt {a x^2+b x+c}}{512 a^3}-\frac {5 \left (b^2-4 a c\right ) (2 a x+b) \left (a x^2+b x+c\right )^{3/2}}{192 a^2}+\frac {(2 a x+b) \left (a x^2+b x+c\right )^{5/2}}{12 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rubi steps
\begin {align*} \int \left (c+b x+a x^2\right )^{5/2} \, dx &=\frac {(b+2 a x) \left (c+b x+a x^2\right )^{5/2}}{12 a}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \left (c+b x+a x^2\right )^{3/2} \, dx}{24 a}\\ &=-\frac {5 \left (b^2-4 a c\right ) (b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{192 a^2}+\frac {(b+2 a x) \left (c+b x+a x^2\right )^{5/2}}{12 a}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int \sqrt {c+b x+a x^2} \, dx}{128 a^2}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (b+2 a x) \sqrt {c+b x+a x^2}}{512 a^3}-\frac {5 \left (b^2-4 a c\right ) (b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{192 a^2}+\frac {(b+2 a x) \left (c+b x+a x^2\right )^{5/2}}{12 a}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{1024 a^3}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (b+2 a x) \sqrt {c+b x+a x^2}}{512 a^3}-\frac {5 \left (b^2-4 a c\right ) (b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{192 a^2}+\frac {(b+2 a x) \left (c+b x+a x^2\right )^{5/2}}{12 a}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {b+2 a x}{\sqrt {c+b x+a x^2}}\right )}{512 a^3}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (b+2 a x) \sqrt {c+b x+a x^2}}{512 a^3}-\frac {5 \left (b^2-4 a c\right ) (b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{192 a^2}+\frac {(b+2 a x) \left (c+b x+a x^2\right )^{5/2}}{12 a}-\frac {5 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{1024 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 162, normalized size = 0.82 \begin {gather*} \frac {\sqrt {x (a x+b)+c} \left (2 (2 a x+b) \left (32 a^2 b x \left (8 a x^2+13 c\right )+16 a^2 \left (8 a^2 x^4+26 a c x^2+33 c^2\right )-40 a b^3 x+8 a b^2 \left (11 a x^2-20 c\right )+15 b^4\right )+\frac {15 \left (b^2-4 a c\right )^{5/2} \sin ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a (x (a x+b)+c)}{4 a c-b^2}}}\right )}{3072 a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.69, size = 197, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a x^2+b x+c} \left (256 a^5 x^5+640 a^4 b x^4+832 a^4 c x^3+432 a^3 b^2 x^3+1248 a^3 b c x^2+1056 a^3 c^2 x+8 a^2 b^3 x^2+96 a^2 b^2 c x+528 a^2 b c^2-10 a b^4 x-160 a b^3 c+15 b^5\right )}{1536 a^3}-\frac {5 \left (64 a^3 c^3-48 a^2 b^2 c^2+12 a b^4 c-b^6\right ) \log \left (-2 \sqrt {a} \sqrt {a x^2+b x+c}+2 a x+b\right )}{1024 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 425, normalized size = 2.16 \begin {gather*} \left [-\frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - 4 \, \sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {a} - b^{2} - 4 \, a c\right ) - 4 \, {\left (256 \, a^{6} x^{5} + 640 \, a^{5} b x^{4} + 15 \, a b^{5} - 160 \, a^{2} b^{3} c + 528 \, a^{3} b c^{2} + 16 \, {\left (27 \, a^{4} b^{2} + 52 \, a^{5} c\right )} x^{3} + 8 \, {\left (a^{3} b^{3} + 156 \, a^{4} b c\right )} x^{2} - 2 \, {\left (5 \, a^{2} b^{4} - 48 \, a^{3} b^{2} c - 528 \, a^{4} c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{6144 \, a^{4}}, \frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + b x + c} {\left (2 \, a x + b\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 2 \, {\left (256 \, a^{6} x^{5} + 640 \, a^{5} b x^{4} + 15 \, a b^{5} - 160 \, a^{2} b^{3} c + 528 \, a^{3} b c^{2} + 16 \, {\left (27 \, a^{4} b^{2} + 52 \, a^{5} c\right )} x^{3} + 8 \, {\left (a^{3} b^{3} + 156 \, a^{4} b c\right )} x^{2} - 2 \, {\left (5 \, a^{2} b^{4} - 48 \, a^{3} b^{2} c - 528 \, a^{4} c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{3072 \, a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 208, normalized size = 1.06 \begin {gather*} \frac {1}{1536} \, \sqrt {a x^{2} + b x + c} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, a^{2} x + 5 \, a b\right )} x + \frac {27 \, a^{5} b^{2} + 52 \, a^{6} c}{a^{5}}\right )} x + \frac {a^{4} b^{3} + 156 \, a^{5} b c}{a^{5}}\right )} x - \frac {5 \, a^{3} b^{4} - 48 \, a^{4} b^{2} c - 528 \, a^{5} c^{2}}{a^{5}}\right )} x + \frac {15 \, a^{2} b^{5} - 160 \, a^{3} b^{3} c + 528 \, a^{4} b c^{2}}{a^{5}}\right )} + \frac {5 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x + c}\right )} \sqrt {a} - b \right |}\right )}{1024 \, a^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 360, normalized size = 1.83 \begin {gather*} \frac {\left (2 a x +b \right ) \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}}}{12 a}+\frac {5 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} x c}{24}-\frac {5 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} x \,b^{2}}{96 a}+\frac {5 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b c}{48 a}-\frac {5 \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{3}}{192 a^{2}}+\frac {5 \sqrt {a \,x^{2}+b x +c}\, x \,c^{2}}{16}-\frac {5 \sqrt {a \,x^{2}+b x +c}\, x c \,b^{2}}{32 a}+\frac {5 \sqrt {a \,x^{2}+b x +c}\, x \,b^{4}}{256 a^{2}}+\frac {5 \sqrt {a \,x^{2}+b x +c}\, b \,c^{2}}{32 a}-\frac {5 \sqrt {a \,x^{2}+b x +c}\, b^{3} c}{64 a^{2}}+\frac {5 \sqrt {a \,x^{2}+b x +c}\, b^{5}}{512 a^{3}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{3}}{16 \sqrt {a}}-\frac {15 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{2} b^{2}}{64 a^{\frac {3}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{4} c}{256 a^{\frac {5}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{6}}{1024 a^{\frac {7}{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 [B] time = 1.76, size = 143, normalized size = 0.73 \begin {gather*} \frac {\left (\frac {b}{2}+a\,x\right )\,{\left (a\,x^2+b\,x+c\right )}^{5/2}}{6\,a}+\frac {\left (5\,a\,c-\frac {5\,b^2}{4}\right )\,\left (\frac {\left (\left (\frac {x}{2}+\frac {b}{4\,a}\right )\,\sqrt {a\,x^2+b\,x+c}+\frac {\ln \left (\frac {\frac {b}{2}+a\,x}{\sqrt {a}}+\sqrt {a\,x^2+b\,x+c}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,a^{3/2}}\right )\,\left (3\,a\,c-\frac {3\,b^2}{4}\right )}{4\,a}+\frac {\left (\frac {b}{2}+a\,x\right )\,{\left (a\,x^2+b\,x+c\right )}^{3/2}}{4\,a}\right )}{6\,a} \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 \left (a x^{2} + b x + c\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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