Optimal. Leaf size=189 \[ -\frac {5 b^7 (9 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {5 b^5 (b+2 c x) \sqrt {b x+c x^2} (9 b B-16 A c)}{16384 c^5}-\frac {5 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-16 A c)}{6144 c^4}+\frac {b (b+2 c x) \left (b x+c x^2\right )^{5/2} (9 b B-16 A c)}{384 c^3}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]
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Rubi [A] time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {779, 612, 620, 206} \[ \frac {5 b^5 (b+2 c x) \sqrt {b x+c x^2} (9 b B-16 A c)}{16384 c^5}-\frac {5 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (9 b B-16 A c)}{6144 c^4}-\frac {5 b^7 (9 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {b (b+2 c x) \left (b x+c x^2\right )^{5/2} (9 b B-16 A c)}{384 c^3}-\frac {\left (b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rubi steps
\begin {align*} \int x (A+B x) \left (b x+c x^2\right )^{5/2} \, dx &=-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {(b (9 b B-16 A c)) \int \left (b x+c x^2\right )^{5/2} \, dx}{32 c^2}\\ &=\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^3 (9 b B-16 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 b^3 (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {\left (5 b^5 (9 b B-16 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 b^5 (9 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^3 (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^7 (9 b B-16 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 b^5 (9 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^3 (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {\left (5 b^7 (9 b B-16 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{16384 c^5}\\ &=\frac {5 b^5 (9 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^3 (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {b (9 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 b^7 (9 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 182, normalized size = 0.96 \[ \frac {(x (b+c x))^{9/2} \left (9 B (b+c x)^3-\frac {3 (9 b B-16 A c) \left (105 b^{13/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \left (-105 b^6+70 b^5 c x-56 b^4 c^2 x^2+48 b^3 c^3 x^3+4736 b^2 c^4 x^4+7424 b c^5 x^5+3072 c^6 x^6\right )\right )}{14336 c^{9/2} x^{9/2} \sqrt {\frac {c x}{b}+1}}\right )}{72 c (b+c x)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 447, normalized size = 2.37 \[ \left [-\frac {105 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (43008 \, B c^{8} x^{7} + 945 \, B b^{7} c - 1680 \, A b^{6} c^{2} + 3072 \, {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 256 \, {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (43008 \, B c^{8} x^{7} + 945 \, B b^{7} c - 1680 \, A b^{6} c^{2} + 3072 \, {\left (33 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 256 \, {\left (243 \, B b^{2} c^{6} + 464 \, A b c^{7}\right )} x^{5} + 128 \, {\left (3 \, B b^{3} c^{5} + 592 \, A b^{2} c^{6}\right )} x^{4} - 48 \, {\left (9 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} + 56 \, {\left (9 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} - 70 \, {\left (9 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 253, normalized size = 1.34 \[ \frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} x + \frac {33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac {243 \, B b^{2} c^{7} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac {3 \, B b^{3} c^{6} + 592 \, A b^{2} c^{7}}{c^{7}}\right )} x - \frac {3 \, {\left (9 \, B b^{4} c^{5} - 16 \, A b^{3} c^{6}\right )}}{c^{7}}\right )} x + \frac {7 \, {\left (9 \, B b^{5} c^{4} - 16 \, A b^{4} c^{5}\right )}}{c^{7}}\right )} x - \frac {35 \, {\left (9 \, B b^{6} c^{3} - 16 \, A b^{5} c^{4}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (9 \, B b^{7} c^{2} - 16 \, A b^{6} c^{3}\right )}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, B b^{8} - 16 \, A b^{7} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 365, normalized size = 1.93 \[ \frac {5 A \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {9}{2}}}-\frac {45 B \,b^{8} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{5} x}{512 c^{3}}+\frac {45 \sqrt {c \,x^{2}+b x}\, B \,b^{6} x}{8192 c^{4}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{6}}{1024 c^{4}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} x}{192 c^{2}}+\frac {45 \sqrt {c \,x^{2}+b x}\, B \,b^{7}}{16384 c^{5}}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} x}{1024 c^{3}}+\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{4}}{384 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b x}{12 c}-\frac {15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{5}}{2048 c^{4}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} x}{64 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,b^{2}}{24 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{3}}{128 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} B x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}} A}{7 c}-\frac {9 \left (c \,x^{2}+b x \right )^{\frac {7}{2}} B b}{112 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 362, normalized size = 1.92 \[ \frac {45 \, \sqrt {c x^{2} + b x} B b^{6} x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} x}{1024 \, c^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{5} x}{512 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} x}{64 \, c^{2}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3} x}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} B x}{8 \, c} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b x}{12 \, c} - \frac {45 \, B b^{8} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, A b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {9}{2}}} + \frac {45 \, \sqrt {c x^{2} + b x} B b^{7}}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5}}{2048 \, c^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{6}}{1024 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3}}{128 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{4}}{384 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} B b}{112 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{2}}{24 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} A}{7 \, c} \]
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 \[ \int x\,{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]
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 \[ \int x \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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