Optimal. Leaf size=197 \[ \frac {\sqrt {c+b x+a x^2} \left (15 b^5-160 a b^3 c+528 a^2 b c^2-10 a b^4 x+96 a^2 b^2 c x+1056 a^3 c^2 x+8 a^2 b^3 x^2+1248 a^3 b c x^2+432 a^3 b^2 x^3+832 a^4 c x^3+640 a^4 b x^4+256 a^5 x^5\right )}{1536 a^3}-\frac {5 \left (-b^6+12 a b^4 c-48 a^2 b^2 c^2+64 a^3 c^3\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{1024 a^{7/2}} \]
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Rubi [A]
time = 0.04, 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 = {626, 635, 212}
\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 212
Rule 626
Rule 635
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 ) \text {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 = 0.58, size = 147, normalized size = 0.75 \begin {gather*} \frac {2 \sqrt {a} (b+2 a x) \sqrt {c+x (b+a x)} \left (15 b^4-40 a b^3 x+32 a^2 b x \left (13 c+8 a x^2\right )+8 a b^2 \left (-20 c+11 a x^2\right )+16 a^2 \left (33 c^2+26 a c x^2+8 a^2 x^4\right )\right )+15 \left (b^2-4 a c\right )^3 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )}{3072 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 143, normalized size = 0.73
method | result | size |
default | \(\frac {\left (2 a x +b \right ) \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}}}{12 a}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 a x +b \right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 a x +b \right ) \sqrt {a \,x^{2}+b x +c}}{4 a}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{24 a}\) | \(143\) |
risch | \(\frac {\sqrt {a \,x^{2}+b x +c}\, \left (256 a^{5} x^{5}+640 a^{4} b \,x^{4}+832 a^{4} c \,x^{3}+432 a^{3} b^{2} x^{3}+1248 a^{3} b c \,x^{2}+8 a^{2} b^{3} x^{2}+1056 a^{3} c^{2} x +96 a^{2} b^{2} c x -10 a \,b^{4} x +528 a^{2} b \,c^{2}-160 a \,b^{3} c +15 b^{5}\right )}{1536 a^{3}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{3}}{16 \sqrt {a}}-\frac {15 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{2} c^{2}}{64 a^{\frac {3}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{4} c}{256 a^{\frac {5}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) b^{6}}{1024 a^{\frac {7}{2}}}\) | \(261\) |
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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Fricas [A]
time = 0.42, 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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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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Giac [A]
time = 0.41, 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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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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