Optimal. Leaf size=158 \[ -\frac {3 \left (b^2-4 a c\right )^2 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}}+\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (b B-2 A c)}{128 c^3}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \[ \frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (b B-2 A c)}{128 c^3}-\frac {3 \left (b^2-4 a c\right )^2 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (b B-2 A c)}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rubi steps
\begin {align*} \int (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {(-b B+2 A c) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{2 c}\\ &=-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c}+\frac {\left (3 \left (b^2-4 a c\right ) (b B-2 A c)\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^2}\\ &=\frac {3 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right )^2 (b B-2 A c)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^3}\\ &=\frac {3 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right )^2 (b B-2 A c)\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^3}\\ &=\frac {3 \left (b^2-4 a c\right ) (b B-2 A c) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3}-\frac {(b B-2 A c) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2}+\frac {B \left (a+b x+c x^2\right )^{5/2}}{5 c}-\frac {3 \left (b^2-4 a c\right )^2 (b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 144, normalized size = 0.91 \[ \frac {5 (2 A c-b B) \left (\frac {3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{128 c^{5/2}}+\frac {(b+2 c x) (a+x (b+c x))^{3/2}}{8 c}\right )+2 B (a+x (b+c x))^{5/2}}{10 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 515, normalized size = 3.26 \[ \left [\frac {15 \, {\left (B b^{5} - 32 \, A a^{2} c^{3} + 16 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (128 \, B c^{5} x^{4} + 15 \, B b^{4} c + 8 \, {\left (16 \, B a^{2} + 25 \, A a b\right )} c^{3} + 16 \, {\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 10 \, {\left (10 \, B a b^{2} + 3 \, A b^{3}\right )} c^{2} + 8 \, {\left (B b^{2} c^{3} + 2 \, {\left (16 \, B a + 15 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{3} c^{2} - 200 \, A a c^{4} - 2 \, {\left (14 \, B a b + 5 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2560 \, c^{4}}, \frac {15 \, {\left (B b^{5} - 32 \, A a^{2} c^{3} + 16 \, {\left (B a^{2} b + A a b^{2}\right )} c^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (128 \, B c^{5} x^{4} + 15 \, B b^{4} c + 8 \, {\left (16 \, B a^{2} + 25 \, A a b\right )} c^{3} + 16 \, {\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 10 \, {\left (10 \, B a b^{2} + 3 \, A b^{3}\right )} c^{2} + 8 \, {\left (B b^{2} c^{3} + 2 \, {\left (16 \, B a + 15 \, A b\right )} c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{3} c^{2} - 200 \, A a c^{4} - 2 \, {\left (14 \, B a b + 5 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1280 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 251, normalized size = 1.59 \[ \frac {1}{640} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, B c x + \frac {11 \, B b c^{4} + 10 \, A c^{5}}{c^{4}}\right )} x + \frac {B b^{2} c^{3} + 32 \, B a c^{4} + 30 \, A b c^{4}}{c^{4}}\right )} x - \frac {5 \, B b^{3} c^{2} - 28 \, B a b c^{3} - 10 \, A b^{2} c^{3} - 200 \, A a c^{4}}{c^{4}}\right )} x + \frac {15 \, B b^{4} c - 100 \, B a b^{2} c^{2} - 30 \, A b^{3} c^{2} + 128 \, B a^{2} c^{3} + 200 \, A a b c^{3}}{c^{4}}\right )} + \frac {3 \, {\left (B b^{5} - 8 \, B a b^{3} c - 2 \, A b^{4} c + 16 \, B a^{2} b c^{2} + 16 \, A a b^{2} c^{2} - 32 \, A a^{2} c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 469, normalized size = 2.97 \[ \frac {3 A \,a^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}-\frac {3 A a \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 A \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}-\frac {3 B \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 B a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}-\frac {3 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a x}{8}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} x}{32 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a b x}{16 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} x}{64 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a b}{16 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A x}{4}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2}}{32 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4}}{128 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b x}{8 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{8 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{16 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{5 c} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 305, normalized size = 1.93 \[ \frac {B\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5\,c}+\frac {A\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )\,\left (3\,a\,c-\frac {3\,b^2}{4}\right )}{4\,c}-\frac {B\,b\,\left (\frac {3\,a\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{4}+\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4}+\frac {b\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{8\,c}-\frac {3\,b^2\,\left (\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (\frac {a}{2\,\sqrt {c}}-\frac {b^2}{8\,c^{3/2}}\right )+\frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{4\,c}\right )}{16\,c}\right )}{2\,c}+\frac {A\,\left (\frac {b}{2}+c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \]
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 \[ \int \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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