Optimal. Leaf size=334 \[ \frac {\sqrt {c+b x+a x^2} \left (-45 b^8+390 a b^6 c+24 a^2 b^4 c^2+160 a^3 b^2 c^3+1920 a^4 c^4+30 a b^7 x+1308 a^2 b^5 c x-80 a^3 b^3 c^2 x-960 a^4 b c^3 x+744 a^2 b^6 x^2+48 a^3 b^4 c x^2+640 a^4 b^2 c^2 x^2+1008 a^3 b^5 x^3-480 a^4 b^3 c x^3+384 a^4 b^4 x^4\right )}{1920 a^2 b^5}-\frac {a^{5/2} c^5 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{b^6}+\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{256 a^{5/2} b^6}+\frac {a^{5/2} c^5 \log \left (2 \sqrt {a} c+\sqrt {a} b x-b \sqrt {c+b x+a x^2}\right )}{b^6} \]
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Rubi [A]
time = 0.25, antiderivative size = 309, normalized size of antiderivative = 0.93, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {748, 828, 857,
635, 212, 738} \begin {gather*} -\frac {a^{5/2} c^5 \tanh ^{-1}\left (\frac {x \left (b^2-2 a c\right )+b c}{2 \sqrt {a} c \sqrt {a x^2+b x+c}}\right )}{b^6}+\frac {\left (16 a^2 c^2+6 a b x \left (b^2-2 a c\right )-6 a b^2 c+3 b^4\right ) \left (a x^2+b x+c\right )^{3/2}}{48 a b^3}+\frac {\left (-256 a^5 c^5+80 a^2 b^6 c^2-30 a b^8 c+3 b^{10}\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{256 a^{5/2} b^6}-\frac {\left (-128 a^4 c^4+32 a^3 b^2 c^3+8 a^2 b^4 c^2+2 a b x \left (b^2-2 a c\right ) \left (-16 a^2 c^2-12 a b^2 c+3 b^4\right )-18 a b^6 c+3 b^8\right ) \sqrt {a x^2+b x+c}}{128 a^2 b^5}+\frac {\left (a x^2+b x+c\right )^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (c+b x+a x^2\right )^{5/2}}{c+b x} \, dx &=\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\int \frac {\left (-b c-\left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{c+b x} \, dx}{2 b}\\ &=\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\int \frac {\left (-\frac {1}{2} b c \left (3 b^4-18 a b^2 c+8 a^2 c^2\right )-\frac {1}{2} \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{c+b x} \, dx}{16 a b^3}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\int \frac {-\frac {1}{4} b^5 c \left (3 b^4-30 a b^2 c+80 a^2 c^2\right )-\frac {1}{4} \left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) x}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{64 a^2 b^5}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\left (a^3 c^6\right ) \int \frac {1}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{b^6}-\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{256 a^2 b^6}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}-\frac {\left (2 a^3 c^6\right ) \text {Subst}\left (\int \frac {1}{4 a c^2-x^2} \, dx,x,\frac {b c-\left (-b^2+2 a c\right ) x}{\sqrt {c+b x+a x^2}}\right )}{b^6}-\frac {\left (-3 b^{10}+30 a b^8 c-80 a^2 b^6 c^2+256 a^5 c^5\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 )}{128 a^2 b^6}\\ &=-\frac {\left (3 b^8-18 a b^6 c+8 a^2 b^4 c^2+32 a^3 b^2 c^3-128 a^4 c^4+2 a b \left (b^2-2 a c\right ) \left (3 b^4-12 a b^2 c-16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{128 a^2 b^5}+\frac {\left (3 b^4-6 a b^2 c+16 a^2 c^2+6 a b \left (b^2-2 a c\right ) x\right ) \left (c+b x+a x^2\right )^{3/2}}{48 a b^3}+\frac {\left (c+b x+a x^2\right )^{5/2}}{5 b}+\frac {\left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{256 a^{5/2} b^6}-\frac {a^{5/2} c^5 \tanh ^{-1}\left (\frac {b c+\left (b^2-2 a c\right ) x}{2 \sqrt {a} c \sqrt {c+b x+a x^2}}\right )}{b^6}\\ \end {align*}
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Mathematica [A]
time = 1.94, size = 295, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {a} b \sqrt {c+x (b+a x)} \left (-45 b^8+1920 a^4 c^4+30 a b^7 x-960 a^4 b c^3 x+160 a^3 b^2 c^2 \left (c+4 a x^2\right )-80 a^3 b^3 c x \left (c+6 a x^2\right )+12 a^2 b^5 x \left (109 c+84 a x^2\right )+6 a b^6 \left (65 c+124 a x^2\right )+24 a^2 b^4 \left (c^2+2 a c x^2+16 a^2 x^4\right )\right )-3840 a^5 c^5 \log \left (-\sqrt {a} x+\sqrt {c+x (b+a x)}\right )-15 \left (3 b^{10}-30 a b^8 c+80 a^2 b^6 c^2-256 a^5 c^5\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )+3840 a^5 c^5 \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+x (b+a x)}\right )}{3840 a^{5/2} b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(880\) vs.
\(2(308)=616\).
time = 0.33, size = 881, normalized size = 2.64
method | result | size |
risch | \(\frac {\sqrt {a \,x^{2}+b x +c}\, \left (384 a^{4} b^{4} x^{4}-480 a^{4} b^{3} c \,x^{3}+1008 a^{3} b^{5} x^{3}+640 a^{4} b^{2} c^{2} x^{2}+48 a^{3} b^{4} c \,x^{2}+744 a^{2} b^{6} x^{2}-960 a^{4} b \,c^{3} x -80 a^{3} b^{3} c^{2} x +1308 a^{2} b^{5} c x +30 a \,b^{7} x +1920 a^{4} c^{4}+160 a^{3} b^{2} c^{3}+24 a^{2} b^{4} c^{2}+390 a \,b^{6} c -45 b^{8}\right )}{1920 a^{2} b^{5}}-\frac {a^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{5}}{b^{6}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{2}}{16 \sqrt {a}}-\frac {15 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c}{128 a^{\frac {3}{2}}}+\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{256 a^{\frac {5}{2}}}-\frac {a^{3} c^{6} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{7} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\) | \(431\) |
default | \(\frac {\frac {\left (a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {5}{2}}}{5}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \left (a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {3}{2}}}{8 a}+\frac {3 \left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{8 a^{\frac {3}{2}}}\right )}{16 a}\right )}{2 b}+\frac {a \,c^{2} \left (\frac {\left (a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}\right )^{\frac {3}{2}}}{3}-\frac {\left (2 a c -b^{2}\right ) \left (\frac {\left (2 a \left (x +\frac {c}{b}\right )-\frac {2 a c -b^{2}}{b}\right ) \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{4 a}+\frac {\left (\frac {4 a^{2} c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right )^{2}}{b^{2}}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{8 a^{\frac {3}{2}}}\right )}{2 b}+\frac {a \,c^{2} \left (\sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}-\frac {\left (2 a c -b^{2}\right ) \ln \left (\frac {-\frac {2 a c -b^{2}}{2 b}+a \left (x +\frac {c}{b}\right )}{\sqrt {a}}+\sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}\right )}{2 b \sqrt {a}}-\frac {a \,c^{2} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{2} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\right )}{b^{2}}\right )}{b^{2}}}{b}\) | \(881\) |
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 = 49.06, size = 707, normalized size = 2.12 \begin {gather*} \left [\frac {3840 \, a^{\frac {11}{2}} c^{5} \log \left (-\frac {2 \, b^{3} c x + b^{2} c^{2} + 4 \, a c^{3} + {\left (b^{4} - 4 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{2} - 4 \, {\left (b c^{2} + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )} \sqrt {a x^{2} + b x + c} \sqrt {a}}{b^{2} x^{2} + 2 \, b c x + c^{2}}\right ) - 15 \, {\left (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{7680 \, a^{3} b^{6}}, -\frac {3840 \, \sqrt {-a} a^{5} c^{5} \arctan \left (-\frac {\sqrt {a x^{2} + b x + c} {\left (b c + {\left (b^{2} - 2 \, a c\right )} x\right )} \sqrt {-a}}{2 \, {\left (a^{2} c x^{2} + a b c x + a c^{2}\right )}}\right ) + 15 \, {\left (3 \, b^{10} - 30 \, a b^{8} c + 80 \, a^{2} b^{6} c^{2} - 256 \, a^{5} c^{5}\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 (384 \, a^{5} b^{5} x^{4} - 45 \, a b^{9} + 390 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 160 \, a^{4} b^{3} c^{3} + 1920 \, a^{5} b c^{4} + 48 \, {\left (21 \, a^{4} b^{6} - 10 \, a^{5} b^{4} c\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{7} + 6 \, a^{4} b^{5} c + 80 \, a^{5} b^{3} c^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{2} b^{8} + 654 \, a^{3} b^{6} c - 40 \, a^{4} b^{4} c^{2} - 480 \, a^{5} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{3840 \, a^{3} b^{6}}\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 \frac {\left (a x^{2} + b x + c\right )^{\frac {5}{2}}}{b x + c}\, dx \end {gather*}
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^2+b\,x+c\right )}^{5/2}}{c+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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