Optimal. Leaf size=147 \[ \frac {5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{7/2} d^3}-\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{64 c^3 d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 147, 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 = {684, 685, 688, 205} \begin {gather*} -\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{64 c^3 d^3}+\frac {5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{7/2} d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 684
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^3} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx}{8 c d^2}\\ &=\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx}{32 c^2 d^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{64 c^3 d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{128 c^3 d^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{64 c^3 d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \operatorname {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 )}{32 c^2 d^2}\\ &=-\frac {5 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{64 c^3 d^3}+\frac {5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2}+\frac {5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{128 c^{7/2} d^3}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 0.42 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{7 d^3 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.48, size = 195, normalized size = 1.33 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-48 a^2 c^2+80 a b^2 c+224 a b c^2 x+224 a c^3 x^2-15 b^4-40 b^3 c x-8 b^2 c^2 x^2+64 b c^3 x^3+32 c^4 x^4\right )}{192 c^3 d^3 (b+2 c x)^2}-\frac {5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (-\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+\frac {2 c x}{\sqrt {b^2-4 a c}}+\frac {b}{\sqrt {b^2-4 a c}}\right )}{64 c^{7/2} d^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 480, normalized size = 3.27 \begin {gather*} \left [-\frac {15 \, {\left (b^{4} - 4 \, a b^{2} c + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{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} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, {\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \, {\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \, {\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}, -\frac {15 \, {\left (b^{4} - 4 \, a b^{2} c + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (\frac {\sqrt {\frac {b^{2} - 4 \, a c}{c}}}{2 \, \sqrt {c x^{2} + b x + a}}\right ) - 2 \, {\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \, {\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \, {\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 519, normalized size = 3.53 \begin {gather*} \frac {1}{48} \, \sqrt {c x^{2} + b x + a} {\left (2 \, x {\left (\frac {x}{c d^{3}} + \frac {b}{c^{2} d^{3}}\right )} - \frac {3 \, b^{2} c^{5} d^{9} - 14 \, a c^{6} d^{9}}{c^{8} d^{12}}\right )} + \frac {5 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{64 \, \sqrt {b^{2} c - 4 \, a c^{2}} c^{3} d^{3}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{4} c^{\frac {3}{2}} - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{2} c^{\frac {5}{2}} + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} c^{\frac {7}{2}} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{5} c - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{3} c^{2} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b c^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{6} \sqrt {c} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{4} c^{\frac {3}{2}} + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{3} c^{\frac {7}{2}} + a b^{5} c - 8 \, a^{2} b^{3} c^{2} + 16 \, a^{3} b c^{3}}{64 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {3}{2}} + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c + b^{2} \sqrt {c} - 2 \, a c^{\frac {3}{2}}\right )}^{2} c^{\frac {5}{2}} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 840, normalized size = 5.71 \begin {gather*} -\frac {5 a^{3} \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 )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{3}}+\frac {15 a^{2} b^{2} \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 )}{8 \left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{3}}-\frac {15 a \,b^{4} \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 )}{32 \left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{3}}+\frac {5 b^{6} \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 )}{128 \left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{4} d^{3}}+\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a^{2}}{8 \left (4 a c -b^{2}\right ) c \,d^{3}}-\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a \,b^{2}}{16 \left (4 a c -b^{2}\right ) c^{2} d^{3}}+\frac {5 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{4}}{128 \left (4 a c -b^{2}\right ) c^{3} d^{3}}+\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a}{12 \left (4 a c -b^{2}\right ) c \,d^{3}}-\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2}}{48 \left (4 a c -b^{2}\right ) c^{2} d^{3}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{4 \left (4 a c -b^{2}\right ) c \,d^{3}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{4 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{3}} \end {gather*}
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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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 )}^3} \,d x \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^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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