Optimal. Leaf size=281 \[ \frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac {\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac {\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac {x^3 \sqrt {a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^3 \left (-4 a B-\frac {1}{2} (9 b B-10 A c) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^2 \left (\frac {3}{2} a (9 b B-10 A c)+\frac {1}{4} \left (63 b^2 B-70 A b c-64 a B c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x \left (-\frac {1}{2} a \left (63 b^2 B-70 A b c-64 a B c\right )-\frac {1}{8} \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^5}\\ &=\frac {\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {(9 b B-10 A c) x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {B x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 225, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (16 c^2 \left (64 a^2 B-a c x (45 A+32 B x)+6 c^2 x^3 (5 A+4 B x)\right )+28 b^2 c (c x (25 A+18 B x)-105 a B)+8 b c^2 \left (a (275 A+161 B x)-2 c x^2 (35 A+27 B x)\right )-210 b^3 c (5 A+3 B x)+945 b^4 B\right )}{1920 c^5}+\frac {\left (96 a^2 A c^3-240 a^2 b B c^2-240 a A b^2 c^2+280 a b^3 B c+70 A b^4 c-63 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{256 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.92, size = 244, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (1024 a^2 B c^2+2200 a A b c^2-720 a A c^3 x-2940 a b^2 B c+1288 a b B c^2 x-512 a B c^3 x^2-1050 A b^3 c+700 A b^2 c^2 x-560 A b c^3 x^2+480 A c^4 x^3+945 b^4 B-630 b^3 B c x+504 b^2 B c^2 x^2-432 b B c^3 x^3+384 B c^4 x^4\right )}{1920 c^5}+\frac {\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{256 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 519, normalized size = 1.85 \begin {gather*} \left [-\frac {15 \, {\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\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 (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {15 \, {\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\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 (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \, {\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \, {\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \, {\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \, {\left (63 \, B b^{2} c^{3} - 2 \, {\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \, {\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 249, normalized size = 0.89 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, B x}{c} - \frac {9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac {63 \, B b^{2} c^{2} - 64 \, B a c^{3} - 70 \, A b c^{3}}{c^{5}}\right )} x - \frac {315 \, B b^{3} c - 644 \, B a b c^{2} - 350 \, A b^{2} c^{2} + 360 \, A a c^{3}}{c^{5}}\right )} x + \frac {945 \, B b^{4} - 2940 \, B a b^{2} c - 1050 \, A b^{3} c + 1024 \, B a^{2} c^{2} + 2200 \, A a b c^{2}}{c^{5}}\right )} + \frac {{\left (63 \, B b^{5} - 280 \, B a b^{3} c - 70 \, A b^{4} c + 240 \, B a^{2} b c^{2} + 240 \, A a b^{2} c^{2} - 96 \, A a^{2} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 531, normalized size = 1.89 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, B \,x^{4}}{5 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,x^{3}}{4 c}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, B b \,x^{3}}{40 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A b \,x^{2}}{24 c^{2}}-\frac {4 \sqrt {c \,x^{2}+b x +a}\, B a \,x^{2}}{15 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} x^{2}}{80 c^{3}}+\frac {3 A \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {15 A a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {15 B \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {35 B a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {9}{2}}}-\frac {63 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a x}{8 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{96 c^{3}}+\frac {161 \sqrt {c \,x^{2}+b x +a}\, B a b x}{240 c^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} x}{64 c^{4}}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, A a b}{48 c^{3}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c^{4}}+\frac {8 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2}}{15 c^{3}}-\frac {49 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2}}{32 c^{4}}+\frac {63 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{128 c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (A + B x\right )}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________