Optimal. Leaf size=165 \[ -\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right )^3 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 165, 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 = {699, 706, 635,
212} \begin {gather*} -\frac {d^4 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}-\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 699
Rule 706
Rubi steps
\begin {align*} \int (b d+2 c d x)^4 \sqrt {a+b x+c x^2} \, dx &=\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx}{24 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^2 d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{32 c}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^3 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{64 c}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (\left (b^2-4 a c\right )^3 d^4\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 )}{32 c}\\ &=-\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{32 c}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{48 c}+\frac {d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right )^3 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 147, normalized size = 0.89 \begin {gather*} d^4 \left (\frac {(b+2 c x) \sqrt {a+x (b+c x)} \left (3 b^4+56 b^3 c x+32 b c^2 x \left (a+8 c x^2\right )+8 b^2 c \left (4 a+23 c x^2\right )+16 c^2 \left (-3 a^2+2 a c x^2+8 c^2 x^4\right )\right )}{96 c}+\frac {\left (b^2-4 a c\right )^3 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{64 c^{3/2}}\right ) \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. \(1193\) vs.
\(2(143)=286\).
time = 0.70, size = 1194, normalized size = 7.24
method | result | size |
risch | \(-\frac {\left (-256 c^{5} x^{5}-640 b \,c^{4} x^{4}-64 a \,c^{4} x^{3}-624 b^{2} c^{3} x^{3}-96 a b \,c^{3} x^{2}-296 b^{3} c^{2} x^{2}+96 a^{2} c^{3} x -96 b^{2} c^{2} a x -62 c \,b^{4} x +48 a^{2} b \,c^{2}-32 a \,b^{3} c -3 b^{5}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{96 c}+\left (c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3}-\frac {3 \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2}}{4}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4}}{16 \sqrt {c}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6}}{64 c^{\frac {3}{2}}}\right ) d^{4}\) | \(268\) |
default | \(\text {Expression too large to display}\) | \(1194\) |
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 = 1.77, size = 473, normalized size = 2.87 \begin {gather*} \left [-\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} d^{4} \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 (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \, {\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \, {\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \, {\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x + {\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{2}}, \frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} d^{4} \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 (256 \, c^{6} d^{4} x^{5} + 640 \, b c^{5} d^{4} x^{4} + 16 \, {\left (39 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{3} + 8 \, {\left (37 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{2} + 2 \, {\left (31 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} d^{4} x + {\left (3 \, b^{5} c + 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{192 \, c^{2}}\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*} d^{4} \left (\int b^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 16 c^{4} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 32 b c^{3} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 24 b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 b^{3} c x \sqrt {a + b x + c x^{2}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 259, normalized size = 1.57 \begin {gather*} \frac {1}{96} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, c^{4} d^{4} x + 5 \, b c^{3} d^{4}\right )} x + \frac {39 \, b^{2} c^{7} d^{4} + 4 \, a c^{8} d^{4}}{c^{5}}\right )} x + \frac {37 \, b^{3} c^{6} d^{4} + 12 \, a b c^{7} d^{4}}{c^{5}}\right )} x + \frac {31 \, b^{4} c^{5} d^{4} + 48 \, a b^{2} c^{6} d^{4} - 48 \, a^{2} c^{7} d^{4}}{c^{5}}\right )} x + \frac {3 \, b^{5} c^{4} d^{4} + 32 \, a b^{3} c^{5} d^{4} - 48 \, a^{2} b c^{6} d^{4}}{c^{5}}\right )} + \frac {{\left (b^{6} d^{4} - 12 \, a b^{4} c d^{4} + 48 \, a^{2} b^{2} c^{2} d^{4} - 64 \, a^{3} c^{3} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{64 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.19, size = 1144, normalized size = 6.93 \begin {gather*} 8\,a\,c^3\,d^4\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )-12\,b\,c^3\,d^4\,\left (\frac {7\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )+b^4\,d^4\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {8\,c^3\,d^4\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{3}+\frac {112\,b^2\,c^2\,d^4\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{5}-15\,b^3\,c\,d^4\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )+\frac {32\,b\,c^2\,d^4\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5}-6\,a\,b^2\,c\,d^4\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )-\frac {64\,a\,b\,c^2\,d^4\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5}+\frac {b^4\,d^4\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}+\frac {b^3\,d^4\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{2\,c^{3/2}}+6\,b^2\,c\,d^4\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}+\frac {b^3\,d^4\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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