Optimal. Leaf size=153 \[ -\frac {15 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{256 c^3 d^2}+\frac {5 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{32 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 c d^2 (b+2 c x)}+\frac {15 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{512 c^{7/2} d^2} \]
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Rubi [A]
time = 0.04, 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 = {698, 626, 635,
212} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 698
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 ) \text {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 [A]
time = 0.64, size = 152, normalized size = 0.99 \begin {gather*} \frac {\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-15 b^4-20 b^3 c x+4 b^2 c \left (25 a+3 c x^2\right )+16 b c^2 x \left (9 a+4 c x^2\right )+16 c^2 \left (-8 a^2+9 a c x^2+2 c^2 x^4\right )\right )}{b+2 c x}-15 \left (b^2-4 a c\right )^2 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{512 c^{7/2} d^2} \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. \(293\) vs.
\(2(131)=262\).
time = 0.87, size = 294, normalized size = 1.92
method | result | size |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {24 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{6}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{4 a c -b^{2}}}{4 d^{2} c^{2}}\) | \(294\) |
risch | \(\frac {\left (16 c^{3} x^{3}+24 b \,c^{2} x^{2}+72 a \,c^{2} x -6 b^{2} c x +36 a b c -7 b^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{256 c^{3} d^{2}}+\frac {\frac {15 a^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {3}{2}}}-\frac {15 a \,b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}+\frac {15 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {7}{2}}}-\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{3}}{c \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a^{2} b^{2}}{4 c^{2} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}-\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a \,b^{4}}{16 c^{3} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{6}}{64 c^{4} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}}{d^{2}}\) | \(413\) |
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 = 2.29, 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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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 814 vs.
\(2 (131) = 262\).
time = 3.17, size = 814, normalized size = 5.32 \begin {gather*} -\frac {1}{512} \, d^{2} {\left (\frac {15 \, {\left (b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 8 \, a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 16 \, a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )\right )} \arctan \left (\frac {\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{3} d^{4} {\left | c \right |}} + \frac {8 \, {\left (\sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 8 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 16 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )\right )}}{c^{4} d^{4} {\left | c \right |}} - \frac {9 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} b^{4} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 72 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} a b^{2} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 7 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} b^{4} c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 144 \, {\left (-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c\right )}^{\frac {3}{2}} a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) + 56 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a b^{2} c^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right ) - 112 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} a^{2} c^{3} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{{\left (\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}}\right )}^{2} c^{3} d^{4} {\left | c \right |}}\right )} {\left | c \right |} \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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