Optimal. Leaf size=249 \[ \frac {\sqrt {d+e x} \sqrt {e+f x} \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e f^3 \left (e^2-d f\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right ) \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e^{3/2} f^{7/2}}+\frac {2 (d+e x)^{3/2} \left (a+\frac {e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2} \]
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Rubi [A] time = 0.28, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {949, 80, 50, 63, 217, 206} \[ \frac {\sqrt {d+e x} \sqrt {e+f x} \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e f^3 \left (e^2-d f\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right ) \left (4 e f \left (-2 a e f-b d f+3 b e^2\right )-c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{4 e^{3/2} f^{7/2}}+\frac {2 (d+e x)^{3/2} \left (a+\frac {e (c e-b f)}{f^2}\right )}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rule 949
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x} \left (a+b x+c x^2\right )}{(e+f x)^{3/2}} \, dx &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {f \left (3 b e^2-b d f-2 a e f\right )-c \left (3 e^3-d e f\right )}{2 f^2}-\frac {1}{2} c \left (d-\frac {e^2}{f}\right ) x\right )}{\sqrt {e+f x}} \, dx}{e^2-d f}\\ &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {e+f x}} \, dx}{4 e f^2 \left (e^2-d f\right )}\\ &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {e+f x}} \, dx}{8 e f^3}\\ &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e-\frac {d f}{e}+\frac {f x^2}{e}}} \, dx,x,\sqrt {d+e x}\right )}{4 e^2 f^3}\\ &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {f x^2}{e}} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {e+f x}}\right )}{4 e^2 f^3}\\ &=\frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right )}{4 e^{3/2} f^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 196, normalized size = 0.79 \[ \frac {\frac {\sqrt {e^2-d f} \sqrt {\frac {e (e+f x)}{e^2-d f}} \sinh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e^2-d f}}\right ) \left (4 e f \left (2 a e f+b d f-3 b e^2\right )+c \left (-d^2 f^2-6 d e^2 f+15 e^4\right )\right )}{e}+\sqrt {f} \sqrt {d+e x} \left (4 e f (-2 a f+3 b e+b f x)+c \left (e f \left (d+2 f x^2\right )+d f^2 x-15 e^3-5 e^2 f x\right )\right )}{4 e f^{7/2} \sqrt {e+f x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.23, size = 580, normalized size = 2.33 \[ \left [\frac {{\left (15 \, c e^{5} - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \, {\left (c d e^{3} + 2 \, b e^{4}\right )} f + {\left (15 \, c e^{4} f - {\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \sqrt {e f} \log \left (8 \, e^{2} f^{2} x^{2} + e^{4} + 6 \, d e^{2} f + d^{2} f^{2} + 4 \, {\left (2 \, e f x + e^{2} + d f\right )} \sqrt {e f} \sqrt {e x + d} \sqrt {f x + e} + 8 \, {\left (e^{3} f + d e f^{2}\right )} x\right ) + 4 \, {\left (2 \, c e^{2} f^{3} x^{2} - 15 \, c e^{4} f - 8 \, a e^{2} f^{3} + {\left (c d e^{2} + 12 \, b e^{3}\right )} f^{2} - {\left (5 \, c e^{3} f^{2} - {\left (c d e + 4 \, b e^{2}\right )} f^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {f x + e}}{16 \, {\left (e^{2} f^{5} x + e^{3} f^{4}\right )}}, -\frac {{\left (15 \, c e^{5} - {\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f^{2} - 6 \, {\left (c d e^{3} + 2 \, b e^{4}\right )} f + {\left (15 \, c e^{4} f - {\left (c d^{2} - 4 \, b d e - 8 \, a e^{2}\right )} f^{3} - 6 \, {\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2}\right )} x\right )} \sqrt {-e f} \arctan \left (\frac {{\left (2 \, e f x + e^{2} + d f\right )} \sqrt {-e f} \sqrt {e x + d} \sqrt {f x + e}}{2 \, {\left (e^{2} f^{2} x^{2} + d e^{2} f + {\left (e^{3} f + d e f^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c e^{2} f^{3} x^{2} - 15 \, c e^{4} f - 8 \, a e^{2} f^{3} + {\left (c d e^{2} + 12 \, b e^{3}\right )} f^{2} - {\left (5 \, c e^{3} f^{2} - {\left (c d e + 4 \, b e^{2}\right )} f^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {f x + e}}{8 \, {\left (e^{2} f^{5} x + e^{3} f^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 237, normalized size = 0.95 \[ \frac {{\left ({\left (x e + d\right )} {\left (\frac {2 \, {\left (x e + d\right )} c e^{\left (-1\right )}}{f} - \frac {{\left (3 \, c d f^{4} e^{2} - 4 \, b f^{4} e^{3} + 5 \, c f^{3} e^{4}\right )} e^{\left (-3\right )}}{f^{5}}\right )} + \frac {{\left (c d^{2} f^{4} e^{2} - 4 \, b d f^{4} e^{3} + 6 \, c d f^{3} e^{4} - 8 \, a f^{4} e^{4} + 12 \, b f^{3} e^{5} - 15 \, c f^{2} e^{6}\right )} e^{\left (-3\right )}}{f^{5}}\right )} \sqrt {x e + d}}{4 \, \sqrt {{\left (x e + d\right )} f e - d f e + e^{3}}} + \frac {{\left (c d^{2} f^{2} - 4 \, b d f^{2} e + 6 \, c d f e^{2} - 8 \, a f^{2} e^{2} + 12 \, b f e^{3} - 15 \, c e^{4}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {x e + d} \sqrt {f} e^{\frac {1}{2}} + \sqrt {{\left (x e + d\right )} f e - d f e + e^{3}} \right |}\right )}{4 \, f^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 834, normalized size = 3.35 \[ \frac {\sqrt {e x +d}\, \left (8 a \,e^{2} f^{3} x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+4 b d e \,f^{3} x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-12 b \,e^{3} f^{2} x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-c \,d^{2} f^{3} x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-6 c d \,e^{2} f^{2} x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+15 c \,e^{4} f x \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+8 a \,e^{3} f^{2} \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+4 b d \,e^{2} f^{2} \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-12 b \,e^{4} f \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-c \,d^{2} e \,f^{2} \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )-6 c d \,e^{3} f \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+15 c \,e^{5} \ln \left (\frac {2 e f x +d f +e^{2}+2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}}{2 \sqrt {e f}}\right )+4 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, c e \,f^{2} x^{2}+8 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, b e \,f^{2} x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, c d \,f^{2} x -10 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, c \,e^{2} f x -16 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, a e \,f^{2}+24 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, b \,e^{2} f +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, c d e f -30 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}\, c \,e^{3}\right )}{8 \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {f x +e}\, e \,f^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\sqrt {d+e\,x}\,\left (c\,x^2+b\,x+a\right )}{{\left (e+f\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {d + e x} \left (a + b x + c x^{2}\right )}{\left (e + f x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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