Optimal. Leaf size=238 \[ \frac {9 b^7 (11 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{13/2}}-\frac {9 b^5 (b+2 c x) \sqrt {b x+c x^2} (11 b B-16 A c)}{16384 c^6}+\frac {3 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-16 A c)}{2048 c^5}-\frac {3 b^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{640 c^4}+\frac {3 b x \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{448 c^3}-\frac {x^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c} \]
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Rubi [A] time = 0.25, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {794, 670, 640, 612, 620, 206} \begin {gather*} -\frac {9 b^5 (b+2 c x) \sqrt {b x+c x^2} (11 b B-16 A c)}{16384 c^6}+\frac {3 b^3 (b+2 c x) \left (b x+c x^2\right )^{3/2} (11 b B-16 A c)}{2048 c^5}-\frac {3 b^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{640 c^4}+\frac {9 b^7 (11 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{13/2}}+\frac {3 b x \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{448 c^3}-\frac {x^2 \left (b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (b x+c x^2\right )^{3/2} \, dx &=\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (3 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \int x^3 \left (b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {(9 b (11 b B-16 A c)) \int x^2 \left (b x+c x^2\right )^{3/2} \, dx}{224 c^2}\\ &=\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (3 b^2 (11 b B-16 A c)\right ) \int x \left (b x+c x^2\right )^{3/2} \, dx}{128 c^3}\\ &=-\frac {3 b^2 (11 b B-16 A c) \left (b x+c x^2\right )^{5/2}}{640 c^4}+\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (3 b^3 (11 b B-16 A c)\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{256 c^4}\\ &=\frac {3 b^3 (11 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {3 b^2 (11 b B-16 A c) \left (b x+c x^2\right )^{5/2}}{640 c^4}+\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (9 b^5 (11 b B-16 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{4096 c^5}\\ &=-\frac {9 b^5 (11 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^6}+\frac {3 b^3 (11 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {3 b^2 (11 b B-16 A c) \left (b x+c x^2\right )^{5/2}}{640 c^4}+\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (9 b^7 (11 b B-16 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{32768 c^6}\\ &=-\frac {9 b^5 (11 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^6}+\frac {3 b^3 (11 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {3 b^2 (11 b B-16 A c) \left (b x+c x^2\right )^{5/2}}{640 c^4}+\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {\left (9 b^7 (11 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^6}\\ &=-\frac {9 b^5 (11 b B-16 A c) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^6}+\frac {3 b^3 (11 b B-16 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {3 b^2 (11 b B-16 A c) \left (b x+c x^2\right )^{5/2}}{640 c^4}+\frac {3 b (11 b B-16 A c) x \left (b x+c x^2\right )^{5/2}}{448 c^3}-\frac {(11 b B-16 A c) x^2 \left (b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (b x+c x^2\right )^{5/2}}{8 c}+\frac {9 b^7 (11 b B-16 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 179, normalized size = 0.75 \begin {gather*} \frac {x^5 \sqrt {x (b+c x)} \left (\frac {11 (11 b B-16 A c) \left (315 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 (315 b^6-210 b^5 c x+168 b^4 c^2 x^2-144 b^3 c^3 x^3+128 b^2 c^4 x^4+6400 b c^5 x^5+5120 c^6 x^6\right )\right )}{71680 c^{11/2} x^{11/2} \sqrt {\frac {c x}{b}+1}}+11 B (b+c x)^2\right )}{88 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 225, normalized size = 0.95 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (5040 A b^6 c-3360 A b^5 c^2 x+2688 A b^4 c^3 x^2-2304 A b^3 c^4 x^3+2048 A b^2 c^5 x^4+102400 A b c^6 x^5+81920 A c^7 x^6-3465 b^7 B+2310 b^6 B c x-1848 b^5 B c^2 x^2+1584 b^4 B c^3 x^3-1408 b^3 B c^4 x^4+1280 b^2 B c^5 x^5+87040 b B c^6 x^6+71680 B c^7 x^7\right )}{573440 c^6}-\frac {9 \left (11 b^8 B-16 A b^7 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{32768 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 446, normalized size = 1.87 \begin {gather*} \left [-\frac {315 \, {\left (11 \, 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 (71680 \, B c^{8} x^{7} - 3465 \, B b^{7} c + 5040 \, A b^{6} c^{2} + 5120 \, {\left (17 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 1280 \, {\left (B b^{2} c^{6} + 80 \, A b c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{3} c^{5} - 16 \, A b^{2} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{1146880 \, c^{7}}, -\frac {315 \, {\left (11 \, 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 (71680 \, B c^{8} x^{7} - 3465 \, B b^{7} c + 5040 \, A b^{6} c^{2} + 5120 \, {\left (17 \, B b c^{7} + 16 \, A c^{8}\right )} x^{6} + 1280 \, {\left (B b^{2} c^{6} + 80 \, A b c^{7}\right )} x^{5} - 128 \, {\left (11 \, B b^{3} c^{5} - 16 \, A b^{2} c^{6}\right )} x^{4} + 144 \, {\left (11 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )} x^{3} - 168 \, {\left (11 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )} x^{2} + 210 \, {\left (11 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{573440 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 249, normalized size = 1.05 \begin {gather*} \frac {1}{573440} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, B c x + \frac {17 \, B b c^{7} + 16 \, A c^{8}}{c^{7}}\right )} x + \frac {B b^{2} c^{6} + 80 \, A b c^{7}}{c^{7}}\right )} x - \frac {11 \, B b^{3} c^{5} - 16 \, A b^{2} c^{6}}{c^{7}}\right )} x + \frac {9 \, {\left (11 \, B b^{4} c^{4} - 16 \, A b^{3} c^{5}\right )}}{c^{7}}\right )} x - \frac {21 \, {\left (11 \, B b^{5} c^{3} - 16 \, A b^{4} c^{4}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (11 \, B b^{6} c^{2} - 16 \, A b^{5} c^{3}\right )}}{c^{7}}\right )} x - \frac {315 \, {\left (11 \, B b^{7} c - 16 \, A b^{6} c^{2}\right )}}{c^{7}}\right )} - \frac {9 \, {\left (11 \, 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 {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 373, normalized size = 1.57 \begin {gather*} -\frac {9 A \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}+\frac {99 B \,b^{8} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {13}{2}}}+\frac {9 \sqrt {c \,x^{2}+b x}\, A \,b^{5} x}{512 c^{4}}-\frac {99 \sqrt {c \,x^{2}+b x}\, B \,b^{6} x}{8192 c^{5}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,x^{3}}{8 c}+\frac {9 \sqrt {c \,x^{2}+b x}\, A \,b^{6}}{1024 c^{5}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,x^{2}}{7 c}-\frac {99 \sqrt {c \,x^{2}+b x}\, B \,b^{7}}{16384 c^{6}}+\frac {33 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{4} x}{1024 c^{4}}-\frac {11 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b \,x^{2}}{112 c^{2}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{4}}{128 c^{4}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A b x}{28 c^{2}}+\frac {33 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{5}}{2048 c^{5}}+\frac {33 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{2} x}{448 c^{3}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} A \,b^{2}}{40 c^{3}}-\frac {33 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B \,b^{3}}{640 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 370, normalized size = 1.55 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B x^{3}}{8 \, c} - \frac {11 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b x^{2}}{112 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A x^{2}}{7 \, c} - \frac {99 \, \sqrt {c x^{2} + b x} B b^{6} x}{8192 \, c^{5}} + \frac {33 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{4} x}{1024 \, c^{4}} + \frac {9 \, \sqrt {c x^{2} + b x} A b^{5} x}{512 \, c^{4}} + \frac {33 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{2} x}{448 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{3} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b x}{28 \, c^{2}} + \frac {99 \, B b^{8} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {13}{2}}} - \frac {9 \, A b^{7} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {99 \, \sqrt {c x^{2} + b x} B b^{7}}{16384 \, c^{6}} + \frac {33 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{5}}{2048 \, c^{5}} + \frac {9 \, \sqrt {c x^{2} + b x} A b^{6}}{1024 \, c^{5}} - \frac {33 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b^{3}}{640 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{4}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} A b^{2}}{40 \, c^{3}} \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 x^3\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right ) \,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 x^{3} \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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