Optimal. Leaf size=271 \[ -\frac {3 b^2 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rubi [A] time = 0.35, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {742, 779, 612, 620, 206} \begin {gather*} -\frac {3 b^2 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac {3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 742
Rule 779
Rubi steps
\begin {align*} \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (\frac {1}{2} d (14 c d-5 b e)+\frac {9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{16 c^3}\\ &=\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}-\frac {\left (3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{1024 c^5}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 312, normalized size = 1.15 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )-16 b^3 c^3 \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+32 b^2 c^4 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )+128 b c^5 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )+256 c^6 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )-\frac {105 b^{7/2} \left (3 b^3 e^3-14 b^2 c d e^2+24 b c^2 d^2 e-16 c^3 d^3\right ) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{35840 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.21, size = 372, normalized size = 1.37 \begin {gather*} \frac {3 \left (3 b^7 e^3-14 b^6 c d e^2+24 b^5 c^2 d^2 e-16 b^4 c^3 d^3\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{2048 c^{11/2}}+\frac {\sqrt {b x+c x^2} \left (315 b^6 e^3-1470 b^5 c d e^2-210 b^5 c e^3 x+2520 b^4 c^2 d^2 e+980 b^4 c^2 d e^2 x+168 b^4 c^2 e^3 x^2-1680 b^3 c^3 d^3-1680 b^3 c^3 d^2 e x-784 b^3 c^3 d e^2 x^2-144 b^3 c^3 e^3 x^3+1120 b^2 c^4 d^3 x+1344 b^2 c^4 d^2 e x^2+672 b^2 c^4 d e^2 x^3+128 b^2 c^4 e^3 x^4+13440 b c^5 d^3 x^2+29568 b c^5 d^2 e x^3+23296 b c^5 d e^2 x^4+6400 b c^5 e^3 x^5+8960 c^6 d^3 x^3+21504 c^6 d^2 e x^4+17920 c^6 d e^2 x^5+5120 c^6 e^3 x^6\right )}{35840 c^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 706, normalized size = 2.61 \begin {gather*} \left [-\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{71680 \, c^{6}}, -\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (5120 \, c^{7} e^{3} x^{6} - 1680 \, b^{3} c^{4} d^{3} + 2520 \, b^{4} c^{3} d^{2} e - 1470 \, b^{5} c^{2} d e^{2} + 315 \, b^{6} c e^{3} + 1280 \, {\left (14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}\right )} x^{5} + 128 \, {\left (168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}\right )} x^{4} + 16 \, {\left (560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}\right )} x^{3} + 56 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )} x^{2} + 70 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{35840 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 365, normalized size = 1.35 \begin {gather*} \frac {1}{35840} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c x e^{3} + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac {7 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 629, normalized size = 2.32 \begin {gather*} -\frac {9 b^{7} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2048 c^{\frac {11}{2}}}+\frac {21 b^{6} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}-\frac {9 b^{5} d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}+\frac {3 b^{4} d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {5}{2}}}+\frac {9 \sqrt {c \,x^{2}+b x}\, b^{5} e^{3} x}{512 c^{4}}-\frac {21 \sqrt {c \,x^{2}+b x}\, b^{4} d \,e^{2} x}{256 c^{3}}+\frac {9 \sqrt {c \,x^{2}+b x}\, b^{3} d^{2} e x}{64 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, b^{2} d^{3} x}{32 c}+\frac {9 \sqrt {c \,x^{2}+b x}\, b^{6} e^{3}}{1024 c^{5}}-\frac {21 \sqrt {c \,x^{2}+b x}\, b^{5} d \,e^{2}}{512 c^{4}}+\frac {9 \sqrt {c \,x^{2}+b x}\, b^{4} d^{2} e}{128 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x}\, b^{3} d^{3}}{64 c^{2}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} e^{3} x}{64 c^{3}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d \,e^{2} x}{32 c^{2}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{2} e x}{8 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} e^{3} x^{2}}{7 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} d^{3} x}{4}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} e^{3}}{128 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3} d \,e^{2}}{64 c^{3}}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2} d^{2} e}{16 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b \,d^{3}}{8 c}-\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b \,e^{3} x}{28 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} d \,e^{2} x}{2 c}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2} e^{3}}{40 c^{3}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b d \,e^{2}}{20 c^{2}}+\frac {3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} d^{2} e}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 624, normalized size = 2.30 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} e^{3} x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{3} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{3} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{3} d^{2} e x}{64 \, c^{2}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2} e x}{8 \, c} - \frac {21 \, \sqrt {c x^{2} + b x} b^{4} d e^{2} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d e^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} d e^{2} x}{2 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{5} e^{3} x}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} e^{3} x}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b e^{3} x}{28 \, c^{2}} + \frac {3 \, b^{4} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {9 \, b^{5} d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} + \frac {21 \, b^{6} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {9 \, b^{7} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{3}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{3}}{8 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{4} d^{2} e}{128 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} e}{16 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} e}{5 \, c} - \frac {21 \, \sqrt {c x^{2} + b x} b^{5} d e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d e^{2}}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d e^{2}}{20 \, c^{2}} + \frac {9 \, \sqrt {c x^{2} + b x} b^{6} e^{3}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e^{3}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e^{3}}{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 {\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^3 \,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 \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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