Optimal. Leaf size=176 \[ -\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac {40 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 176, 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 = {701, 707, 702,
211} \begin {gather*} \frac {40 c^{3/2} \text {ArcTan}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{7/2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}+\frac {40 c}{3 d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {2}{3 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 701
Rule 702
Rule 707
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {(20 c) \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}+\frac {\left (80 c^2\right ) \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac {\left (40 c^2\right ) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac {\left (160 c^3\right ) \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {40 c}{3 \left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}+\frac {80 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}+\frac {40 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^3}\\ \end {align*}
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Mathematica [A]
time = 2.49, size = 169, normalized size = 0.96 \begin {gather*} \frac {\frac {-2 b^4+40 b^3 c x+160 b c^2 x \left (2 a+3 c x^2\right )+56 b^2 c \left (a+5 c x^2\right )+16 c^2 \left (3 a^2+20 a c x^2+15 c^2 x^4\right )}{3 \left (b^2-4 a c\right )^3 (b+2 c x)^2 (a+x (b+c x))^{3/2}}-\frac {80 c^{3/2} \tan ^{-1}\left (\frac {b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 296, normalized size = 1.68
method | result | size |
default | \(\frac {-\frac {2 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}-\frac {10 c^{2} \left (\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}+\frac {4 c \left (\frac {4 c}{\left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {8 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}}{8 d^{3} c^{3}}\) | \(296\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 628 vs.
\(2 (156) = 312\).
time = 8.82, size = 1287, normalized size = 7.31 \begin {gather*} \left [-\frac {2 \, {\left (30 \, {\left (4 \, c^{5} x^{6} + 12 \, b c^{4} x^{5} + a^{2} b^{2} c + {\left (13 \, b^{2} c^{3} + 8 \, a c^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c + 10 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - {\left (120 \, c^{4} x^{4} + 240 \, b c^{3} x^{3} - b^{4} + 28 \, a b^{2} c + 24 \, a^{2} c^{2} + 20 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 20 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}}, \frac {2 \, {\left (60 \, {\left (4 \, c^{5} x^{6} + 12 \, b c^{4} x^{5} + a^{2} b^{2} c + {\left (13 \, b^{2} c^{3} + 8 \, a c^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} x^{3} + {\left (b^{4} c + 10 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (120 \, c^{4} x^{4} + 240 \, b c^{3} x^{3} - b^{4} + 28 \, a b^{2} c + 24 \, a^{2} c^{2} + 20 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 20 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\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 {1}{a^{2} b^{3} \sqrt {a + b x + c x^{2}} + 6 a^{2} b^{2} c x \sqrt {a + b x + c x^{2}} + 12 a^{2} b c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 8 a^{2} c^{3} x^{3} \sqrt {a + b x + c x^{2}} + 2 a b^{4} x \sqrt {a + b x + c x^{2}} + 14 a b^{3} c x^{2} \sqrt {a + b x + c x^{2}} + 36 a b^{2} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 40 a b c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 16 a c^{4} x^{5} \sqrt {a + b x + c x^{2}} + b^{5} x^{2} \sqrt {a + b x + c x^{2}} + 8 b^{4} c x^{3} \sqrt {a + b x + c x^{2}} + 25 b^{3} c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 38 b^{2} c^{3} x^{5} \sqrt {a + b x + c x^{2}} + 28 b c^{4} x^{6} \sqrt {a + b x + c x^{2}} + 8 c^{5} x^{7} \sqrt {a + b x + c x^{2}}}\, dx}{d^{3}} \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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