Optimal. Leaf size=305 \[ \frac {\sqrt {b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (-105 b^3 e^3+360 b^2 c d e^2-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c} \]
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Rubi [A] time = 0.44, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {832, 779, 620, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (360 b^2 c d e^2-105 b^3 e^3-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (144 b^2 c^2 d e (A e+B d)-40 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx &=\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (-\frac {1}{2} (b B-8 A c) d+\frac {1}{2} (6 B c d-7 b B e+8 A c e) x\right )}{\sqrt {b x+c x^2}} \, dx}{4 c}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (-\frac {1}{4} d \left (12 b B c d-48 A c^2 d-7 b^2 B e+8 A b c e\right )+\frac {1}{4} \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{12 c^2}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^4}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^4}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 278, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (8 A c e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+10 b^2 c e^2 (36 d+7 e x)-8 b c^2 e \left (54 d^2+30 d e x+7 e^2 x^2\right )+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{192 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.22, size = 322, normalized size = 1.06 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (120 A b^2 c e^3-432 A b c^2 d e^2-80 A b c^2 e^3 x+576 A c^3 d^2 e+288 A c^3 d e^2 x+64 A c^3 e^3 x^2-105 b^3 B e^3+360 b^2 B c d e^2+70 b^2 B c e^3 x-432 b B c^2 d^2 e-240 b B c^2 d e^2 x-56 b B c^2 e^3 x^2+192 B c^3 d^3+288 B c^3 d^2 e x+192 B c^3 d e^2 x^2+48 B c^3 e^3 x^3\right )}{192 c^4}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (40 A b^3 c e^3-144 A b^2 c^2 d e^2+192 A b c^3 d^2 e-128 A c^4 d^3-35 b^4 B e^3+120 b^3 B c d e^2-144 b^2 B c^2 d^2 e+64 b B c^3 d^3\right )}{128 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 621, normalized size = 2.04 \begin {gather*} \left [\frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{5}}, \frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 311, normalized size = 1.02 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B x e^{3}}{c} + \frac {24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac {144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac {3 \, {\left (64 \, B c^{3} d^{3} - 144 \, B b c^{2} d^{2} e + 192 \, A c^{3} d^{2} e + 120 \, B b^{2} c d e^{2} - 144 \, A b c^{2} d e^{2} - 35 \, B b^{3} e^{3} + 40 \, A b^{2} c e^{3}\right )}}{c^{4}}\right )} + \frac {{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 144 \, A b^{2} c^{2} d e^{2} - 35 \, B b^{4} e^{3} + 40 \, A b^{3} c e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 646, normalized size = 2.12 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, B \,e^{3} x^{3}}{4 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,e^{3} x^{2}}{3 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B b \,e^{3} x^{2}}{24 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, B d \,e^{2} x^{2}}{c}-\frac {5 A \,b^{3} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}+\frac {9 A \,b^{2} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {3 A b \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {A \,d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+\frac {35 B \,b^{4} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}-\frac {15 B \,b^{3} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}+\frac {9 B \,b^{2} d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {B b \,d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A b \,e^{3} x}{12 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A d \,e^{2} x}{2 c}+\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{2} e^{3} x}{96 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x}\, B b d \,e^{2} x}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,d^{2} e x}{2 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{2} e^{3}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x}\, A b d \,e^{2}}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,d^{2} e}{c}-\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{3} e^{3}}{64 c^{4}}+\frac {15 \sqrt {c \,x^{2}+b x}\, B \,b^{2} d \,e^{2}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x}\, B b \,d^{2} e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, B \,d^{3}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 474, normalized size = 1.55 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B e^{3} x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b e^{3} x^{2}}{24 \, c^{2}} + \frac {35 \, \sqrt {c x^{2} + b x} B b^{2} e^{3} x}{96 \, c^{3}} + \frac {A d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} + \frac {35 \, B b^{4} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} B b^{3} e^{3}}{64 \, c^{4}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} x^{2}}{3 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b x}{12 \, c^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} x}{2 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{3}} - \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b}{4 \, c^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x}}{c} \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 {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x}} \,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 {\left (A + B x\right ) \left (d + e x\right )^{3}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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