Optimal. Leaf size=362 \[ \frac {\sqrt {c+b x+a x^2} \left (15 b^6 c+28 a b^4 c^2+400 a^2 b^2 c^3-960 a^3 c^4+15 b^7 x+146 a b^5 c x+200 a^2 b^3 c^2 x-480 a^3 b c^3 x+118 a b^6 x^2-64 a^2 b^4 c x^2+160 a^3 b^2 c^2 x^2+136 a^2 b^5 x^3-80 a^3 b^3 c x^3+48 a^3 b^4 x^4\right )}{192 a b^5 (c+b x)}+\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}\right )}{2 b^6}-\frac {5 \left (-b^8+8 a b^6 c-64 a^3 b^2 c^3+128 a^4 c^4\right ) \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+b x+a x^2}\right )}{128 a^{3/2} b^6}-\frac {5 \left (-a^{3/2} b^2 c^3+2 a^{5/2} c^4\right ) \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+b x+a x^2}\right )}{2 b^6} \]
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Rubi [A]
time = 0.24, antiderivative size = 277, normalized size of antiderivative = 0.77, 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 = {746, 828, 857,
635, 212, 738} \begin {gather*} -\frac {5 a^{3/2} c^3 \left (b^2-2 a c\right ) \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 )}{2 b^6}+\frac {5 \left (-64 a^3 c^3+48 a^2 b^2 c^2+2 a b x \left (16 a^2 c^2-4 a b^2 c+b^4\right )-4 a b^4 c+b^6\right ) \sqrt {a x^2+b x+c}}{64 a b^5}-\frac {5 \left (-128 a^4 c^4+64 a^3 b^2 c^3-8 a b^6 c+b^8\right ) \tanh ^{-1}\left (\frac {2 a x+b}{2 \sqrt {a} \sqrt {a x^2+b x+c}}\right )}{128 a^{3/2} b^6}+\frac {5 \left (6 a b x-8 a c+7 b^2\right ) \left (a x^2+b x+c\right )^{3/2}}{24 b^3}-\frac {\left (a x^2+b x+c\right )^{5/2}}{b (b x+c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 828
Rule 857
Rubi steps
\begin {align*} \int \frac {\left (c+b x+a x^2\right )^{5/2}}{(c+b x)^2} \, dx &=-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {5 \int \frac {(b+2 a x) \left (c+b x+a x^2\right )^{3/2}}{c+b x} \, dx}{2 b}\\ &=\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {5 \int \frac {\left (-a b c \left (b^2+4 a c\right )-a \left (b^4-4 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 {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {5 \int \frac {-\frac {1}{2} a b^5 c \left (b^2-8 a c\right )-\frac {1}{2} a \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) x}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{64 a^2 b^5}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}+\frac {\left (5 a^2 c^4 \left (b^2-2 a c\right )\right ) \int \frac {1}{(c+b x) \sqrt {c+b x+a x^2}} \, dx}{2 b^6}-\frac {\left (5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right )\right ) \int \frac {1}{\sqrt {c+b x+a x^2}} \, dx}{128 a b^6}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {\left (5 a^2 c^4 \left (b^2-2 a c\right )\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 (5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right )\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 )}{64 a b^6}\\ &=\frac {5 \left (b^6-4 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3+2 a b \left (b^4-4 a b^2 c+16 a^2 c^2\right ) x\right ) \sqrt {c+b x+a x^2}}{64 a b^5}+\frac {5 \left (7 b^2-8 a c+6 a b x\right ) \left (c+b x+a x^2\right )^{3/2}}{24 b^3}-\frac {\left (c+b x+a x^2\right )^{5/2}}{b (c+b x)}-\frac {5 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) \tanh ^{-1}\left (\frac {b+2 a x}{2 \sqrt {a} \sqrt {c+b x+a x^2}}\right )}{128 a^{3/2} b^6}-\frac {5 a^{3/2} c^3 \left (b^2-2 a c\right ) \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 )}{2 b^6}\\ \end {align*}
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Mathematica [A]
time = 1.38, size = 316, normalized size = 0.87 \begin {gather*} \frac {\frac {2 \sqrt {a} b \sqrt {c+x (b+a x)} \left (-960 a^3 c^4+15 b^7 x-480 a^3 b c^3 x-40 a^2 b^3 c x \left (-5 c+2 a x^2\right )+80 a^2 b^2 c^2 \left (5 c+2 a x^2\right )+2 a b^5 x \left (73 c+68 a x^2\right )+b^6 \left (15 c+118 a x^2\right )+4 a b^4 \left (7 c^2-16 a c x^2+12 a^2 x^4\right )\right )}{c+b x}+960 a^3 c^3 \left (-b^2+2 a c\right ) \log \left (-\sqrt {a} x+\sqrt {c+x (b+a x)}\right )+15 \left (b^8-8 a b^6 c+64 a^3 b^2 c^3-128 a^4 c^4\right ) \log \left (a \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )-960 a^3 c^3 \left (-b^2+2 a c\right ) \log \left (\sqrt {a} (2 c+b x)-b \sqrt {c+x (b+a x)}\right )}{384 a^{3/2} 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. \(1377\) vs.
\(2(330)=660\).
time = 0.11, size = 1378, normalized size = 3.81
method | result | size |
risch | \(\frac {\left (48 a^{3} x^{3} b^{3}-128 a^{3} b^{2} c \,x^{2}+136 a^{2} b^{4} x^{2}+288 a^{3} b \,c^{2} x -200 a^{2} b^{3} c x +118 a \,b^{5} x -768 a^{3} c^{3}+400 a^{2} b^{2} c^{2}+28 a \,b^{4} c +15 b^{6}\right ) \sqrt {a \,x^{2}+b x +c}}{192 a \,b^{5}}+\frac {5 a^{\frac {5}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{4}}{b^{6}}-\frac {5 a^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c^{3}}{2 b^{4}}+\frac {5 \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right ) c}{16 \sqrt {a}}-\frac {5 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{128 a^{\frac {3}{2}}}-\frac {a^{2} c^{4} \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{b^{6} \left (x +\frac {c}{b}\right )}+\frac {5 a^{3} c^{5} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{b^{7} \sqrt {\frac {a \,c^{2}}{b^{2}}}}-\frac {5 a^{2} c^{4} \ln \left (\frac {\frac {2 a \,c^{2}}{b^{2}}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+2 \sqrt {\frac {a \,c^{2}}{b^{2}}}\, \sqrt {a \left (x +\frac {c}{b}\right )^{2}-\frac {\left (2 a c -b^{2}\right ) \left (x +\frac {c}{b}\right )}{b}+\frac {a \,c^{2}}{b^{2}}}}{x +\frac {c}{b}}\right )}{2 b^{5} \sqrt {\frac {a \,c^{2}}{b^{2}}}}\) | \(555\) |
default | \(\text {Expression too large to display}\) | \(1378\) |
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 = 15.34, size = 854, normalized size = 2.36 \begin {gather*} \left [-\frac {15 \, {\left (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\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 ) + 960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {a} \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 ) - 4 \, {\left (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{768 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}, -\frac {960 \, {\left (a^{3} b^{2} c^{4} - 2 \, a^{4} c^{5} + {\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b c^{4}\right )} x\right )} \sqrt {-a} \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 (b^{8} c - 8 \, a b^{6} c^{2} + 64 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5} + {\left (b^{9} - 8 \, a b^{7} c + 64 \, a^{3} b^{3} c^{3} - 128 \, a^{4} b c^{4}\right )} x\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 (48 \, a^{4} b^{5} x^{4} + 15 \, a b^{7} c + 28 \, a^{2} b^{5} c^{2} + 400 \, a^{3} b^{3} c^{3} - 960 \, a^{4} b c^{4} + 8 \, {\left (17 \, a^{3} b^{6} - 10 \, a^{4} b^{4} c\right )} x^{3} + 2 \, {\left (59 \, a^{2} b^{7} - 32 \, a^{3} b^{5} c + 80 \, a^{4} b^{3} c^{2}\right )} x^{2} + {\left (15 \, a b^{8} + 146 \, a^{2} b^{6} c + 200 \, a^{3} b^{4} c^{2} - 480 \, a^{4} b^{2} c^{3}\right )} x\right )} \sqrt {a x^{2} + b x + c}}{384 \, {\left (a^{2} b^{7} x + a^{2} b^{6} c\right )}}\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}}}{\left (b x + c\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}}{{\left (c+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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