Optimal. Leaf size=227 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a d e+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{128 d^{7/2} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt {c+d x^2+e x} \left (8 a d e+4 b c d-5 b e^2\right )}{64 d^3 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)} \]
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Rubi [A] time = 0.13, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1000, 779, 612, 621, 206} \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 d x+e) \sqrt {c+d x^2+e x} \left (8 a d e+4 b c d-5 b e^2\right )}{64 d^3 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 c d-e^2\right ) \left (8 a d e+4 b c d-5 b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{128 d^{7/2} (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2+e x\right )^{3/2} (8 a d+6 b d x-5 b e)}{24 d^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 1000
Rubi steps
\begin {align*} \int x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {(8 a d-5 b e+6 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{24 d^2 (a+b x)}-\frac {\left (b \left (4 b c d+8 a d e-5 b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+e x+d x^2} \, dx}{8 d^2 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (4 b c d+8 a d e-5 b e^2\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{64 d^3 (a+b x)}+\frac {(8 a d-5 b e+6 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{24 d^2 (a+b x)}-\frac {\left (b \left (4 c d-e^2\right ) \left (4 b c d+8 a d e-5 b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{64 d^3 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (4 b c d+8 a d e-5 b e^2\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{64 d^3 (a+b x)}+\frac {(8 a d-5 b e+6 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{24 d^2 (a+b x)}-\frac {\left (b \left (4 c d-e^2\right ) \left (4 b c d+8 a d e-5 b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{32 d^3 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {\left (4 b c d+8 a d e-5 b e^2\right ) (e+2 d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{64 d^3 (a+b x)}+\frac {(8 a d-5 b e+6 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+e x+d x^2\right )^{3/2}}{24 d^2 (a+b x)}-\frac {\left (4 c d-e^2\right ) \left (4 b c d+8 a d e-5 b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{128 d^{7/2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 147, normalized size = 0.65 \[ \frac {\sqrt {(a+b x)^2} \left ((c+x (d x+e))^{3/2} (8 a d+6 b d x-5 b e)-\frac {3 \left (8 a d e+4 b c d-5 b e^2\right ) \left (\left (4 c d-e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {d} (2 d x+e) \sqrt {c+x (d x+e)}\right )}{16 d^{3/2}}\right )}{24 d^2 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 391, normalized size = 1.72 \[ \left [\frac {3 \, {\left (16 \, b c^{2} d^{2} + 32 \, a c d^{2} e - 24 \, b c d e^{2} - 8 \, a d e^{3} + 5 \, b e^{4}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (48 \, b d^{4} x^{3} + 64 \, a c d^{3} - 52 \, b c d^{2} e - 24 \, a d^{2} e^{2} + 15 \, b d e^{3} + 8 \, {\left (8 \, a d^{4} + b d^{3} e\right )} x^{2} + 2 \, {\left (12 \, b c d^{3} + 8 \, a d^{3} e - 5 \, b d^{2} e^{2}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{768 \, d^{4}}, \frac {3 \, {\left (16 \, b c^{2} d^{2} + 32 \, a c d^{2} e - 24 \, b c d e^{2} - 8 \, a d e^{3} + 5 \, b e^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (48 \, b d^{4} x^{3} + 64 \, a c d^{3} - 52 \, b c d^{2} e - 24 \, a d^{2} e^{2} + 15 \, b d e^{3} + 8 \, {\left (8 \, a d^{4} + b d^{3} e\right )} x^{2} + 2 \, {\left (12 \, b c d^{3} + 8 \, a d^{3} e - 5 \, b d^{2} e^{2}\right )} x\right )} \sqrt {d x^{2} + e x + c}}{384 \, d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 268, normalized size = 1.18 \[ \frac {1}{192} \, \sqrt {d x^{2} + x e + c} {\left (2 \, {\left (4 \, {\left (6 \, b x \mathrm {sgn}\left (b x + a\right ) + \frac {8 \, a d^{3} \mathrm {sgn}\left (b x + a\right ) + b d^{2} e \mathrm {sgn}\left (b x + a\right )}{d^{3}}\right )} x + \frac {12 \, b c d^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a d^{2} e \mathrm {sgn}\left (b x + a\right ) - 5 \, b d e^{2} \mathrm {sgn}\left (b x + a\right )}{d^{3}}\right )} x + \frac {64 \, a c d^{2} \mathrm {sgn}\left (b x + a\right ) - 52 \, b c d e \mathrm {sgn}\left (b x + a\right ) - 24 \, a d e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, b e^{3} \mathrm {sgn}\left (b x + a\right )}{d^{3}}\right )} + \frac {{\left (16 \, b c^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + 32 \, a c d^{2} e \mathrm {sgn}\left (b x + a\right ) - 24 \, b c d e^{2} \mathrm {sgn}\left (b x + a\right ) - 8 \, a d e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, b e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} \sqrt {d} - e \right |}\right )}{128 \, d^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 381, normalized size = 1.68 \[ \frac {\left (-96 a c \,d^{3} e \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+24 a \,d^{2} e^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-48 b \,c^{2} d^{3} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+72 b c \,d^{2} e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-15 b d \,e^{4} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-96 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {7}{2}} e x -48 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {7}{2}} x +60 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {5}{2}} e^{2} x -48 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e^{2}-24 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {5}{2}} e +96 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {7}{2}} x +30 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {3}{2}} e^{3}+128 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {7}{2}}-80 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} b \,d^{\frac {5}{2}} e \right ) \mathrm {csgn}\left (b x +a \right )}{384 d^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}} x\,{d x} \]
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 \[ \int x\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c} \,d x \]
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 \[ \int x \sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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