Optimal. Leaf size=280 \[ \frac {\sqrt {a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{24 c^4 \left (b^2-4 a c\right )}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{9/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {818, 832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (256 a^2 B c^2-2 c x \left (72 a A c^2-116 a b B c-30 A b^2 c+35 b^3 B\right )+312 a A b c^2-460 a b^2 B c-90 A b^3 c+105 b^4 B\right )}{24 c^4 \left (b^2-4 a c\right )}+\frac {x^2 \sqrt {a+b x+c x^2} \left (-16 a B c-6 A b c+7 b^2 B\right )}{3 c^2 \left (b^2-4 a c\right )}-\frac {\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{9/2}}-\frac {2 x^3 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 818
Rule 832
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {x^2 \left (3 a (b B-2 A c)+\frac {1}{2} \left (7 b^2 B-6 A b c-16 a B c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (7 b^2 B-6 A b c-16 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {2 \int \frac {x \left (-a \left (7 b^2 B-6 A b c-16 a B c\right )-\frac {1}{4} \left (35 b^3 B-30 A b^2 c-116 a b B c+72 a A c^2\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (7 b^2 B-6 A b c-16 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {\left (105 b^4 B-90 A b^3 c-460 a b^2 B c+312 a A b c^2+256 a^2 B c^2-2 c \left (35 b^3 B-30 A b^2 c-116 a b B c+72 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{24 c^4 \left (b^2-4 a c\right )}-\frac {\left (35 b^3 B-30 A b^2 c-60 a b B c+24 a A c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^4}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (7 b^2 B-6 A b c-16 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {\left (105 b^4 B-90 A b^3 c-460 a b^2 B c+312 a A b c^2+256 a^2 B c^2-2 c \left (35 b^3 B-30 A b^2 c-116 a b B c+72 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{24 c^4 \left (b^2-4 a c\right )}-\frac {\left (35 b^3 B-30 A b^2 c-60 a b B c+24 a A c^2\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 )}{8 c^4}\\ &=-\frac {2 x^3 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (7 b^2 B-6 A b c-16 a B c\right ) x^2 \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac {\left (105 b^4 B-90 A b^3 c-460 a b^2 B c+312 a A b c^2+256 a^2 B c^2-2 c \left (35 b^3 B-30 A b^2 c-116 a b B c+72 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{24 c^4 \left (b^2-4 a c\right )}-\frac {\left (35 b^3 B-30 A b^2 c-60 a b B c+24 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 300, normalized size = 1.07 \begin {gather*} \frac {2 \sqrt {c} \left (-256 a^3 B c^2+4 a^2 c \left (-2 b c (39 A+61 B x)+4 c^2 x (9 A-8 B x)+115 b^2 B\right )+a \left (10 b^3 c (9 A+53 B x)+4 b^2 c^2 x (43 B x-93 A)-8 b c^3 x^2 (15 A+7 B x)+16 c^4 x^3 (3 A+2 B x)-105 b^4 B\right )+b^2 x \left (5 b^2 c (18 A-7 B x)+2 b c^2 x (15 A+7 B x)-4 c^3 x^2 (3 A+2 B x)-105 b^3 B\right )\right )+3 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{48 c^{9/2} \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.35, size = 324, normalized size = 1.16 \begin {gather*} \frac {-256 a^3 B c^2-312 a^2 A b c^2+144 a^2 A c^3 x+460 a^2 b^2 B c-488 a^2 b B c^2 x-128 a^2 B c^3 x^2+90 a A b^3 c-372 a A b^2 c^2 x-120 a A b c^3 x^2+48 a A c^4 x^3-105 a b^4 B+530 a b^3 B c x+172 a b^2 B c^2 x^2-56 a b B c^3 x^3+32 a B c^4 x^4+90 A b^4 c x+30 A b^3 c^2 x^2-12 A b^2 c^3 x^3-105 b^5 B x-35 b^4 B c x^2+14 b^3 B c^2 x^3-8 b^2 B c^3 x^4}{24 c^4 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}}+\frac {\left (24 a A c^2-60 a b B c-30 A b^2 c+35 b^3 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{16 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 1035, normalized size = 3.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 367, normalized size = 1.31 \begin {gather*} \frac {{\left ({\left (2 \, {\left (\frac {4 \, {\left (B b^{2} c^{3} - 4 \, B a c^{4}\right )} x}{b^{2} c^{4} - 4 \, a c^{5}} - \frac {7 \, B b^{3} c^{2} - 28 \, B a b c^{3} - 6 \, A b^{2} c^{3} + 24 \, A a c^{4}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {35 \, B b^{4} c - 172 \, B a b^{2} c^{2} - 30 \, A b^{3} c^{2} + 128 \, B a^{2} c^{3} + 120 \, A a b c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {105 \, B b^{5} - 530 \, B a b^{3} c - 90 \, A b^{4} c + 488 \, B a^{2} b c^{2} + 372 \, A a b^{2} c^{2} - 144 \, A a^{2} c^{3}}{b^{2} c^{4} - 4 \, a c^{5}}\right )} x + \frac {105 \, B a b^{4} - 460 \, B a^{2} b^{2} c - 90 \, A a b^{3} c + 256 \, B a^{3} c^{2} + 312 \, A a^{2} b c^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}{24 \, \sqrt {c x^{2} + b x + a}} + \frac {{\left (35 \, B b^{3} - 60 \, B a b c - 30 \, A b^{2} c + 24 \, A a c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 800, normalized size = 2.86 \begin {gather*} \frac {B \,x^{4}}{3 \sqrt {c \,x^{2}+b x +a}\, c}-\frac {13 A a \,b^{2} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {15 A \,b^{4} x}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {A \,x^{3}}{2 \sqrt {c \,x^{2}+b x +a}\, c}-\frac {16 B \,a^{2} b x}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {115 B a \,b^{3} x}{12 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {35 B \,b^{5} x}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {7 B b \,x^{3}}{12 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {13 A a \,b^{3}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {15 A \,b^{5}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {5 A b \,x^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {8 B \,a^{2} b^{2}}{3 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {115 B a \,b^{4}}{24 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {4 B a \,x^{2}}{3 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {35 B \,b^{6}}{32 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{5}}+\frac {35 B \,b^{2} x^{2}}{24 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {3 A a x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {15 A \,b^{2} x}{8 \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {15 B a b x}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {35 B \,b^{3} x}{16 \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {3 A a \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}+\frac {15 A \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {7}{2}}}+\frac {15 B a b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {7}{2}}}-\frac {35 B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {9}{2}}}-\frac {13 A a b}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {15 A \,b^{3}}{16 \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {8 B \,a^{2}}{3 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {115 B a \,b^{2}}{24 \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {35 B \,b^{4}}{32 \sqrt {c \,x^{2}+b x +a}\, c^{5}} \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 = -0.00 \begin {gather*} \int \frac {x^4\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,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*} \int \frac {x^{4} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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