Optimal. Leaf size=249 \[ \frac {2 \left (a+\frac {e (c e-b f)}{f^2}\right ) (d+e x)^{3/2}}{\left (e^2-d f\right ) \sqrt {e+f x}}+\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \sqrt {d+e x} \sqrt {e+f x}}{4 e f^3 \left (e^2-d f\right )}+\frac {c (d+e x)^{3/2} \sqrt {e+f x}}{2 e f^2}-\frac {\left (4 e f \left (3 b e^2-b d f-2 a e f\right )-c \left (15 e^4-6 d e^2 f-d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {e} \sqrt {e+f x}}\right )}{4 e^{3/2} f^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, 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 = {963, 81, 52, 65,
223, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rule 963
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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.44, size = 167, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d+e x} \left (4 e f (3 b e-2 a f+b f x)+c \left (-15 e^3-5 e^2 f x+d f^2 x+e f \left (d+2 f x^2\right )\right )\right )}{4 e f^3 \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 ) \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 {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(833\) vs.
\(2(219)=438\).
time = 0.09, size = 834, normalized size = 3.35
method | result | size |
default | \(\frac {\sqrt {e x +d}\, \left (8 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) a \,e^{2} f^{3} x +4 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b d e \,f^{3} x -12 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b \,e^{3} f^{2} x -\ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,d^{2} f^{3} x -6 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c d \,e^{2} f^{2} x +15 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,e^{4} f x +4 c e \,f^{2} x^{2} \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}+8 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) a \,e^{3} f^{2}+4 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b d \,e^{2} f^{2}-12 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) b \,e^{4} f -\ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,d^{2} e \,f^{2}-6 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c d \,e^{3} f +15 \ln \left (\frac {2 e f x +2 \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, \sqrt {e f}+d f +e^{2}}{2 \sqrt {e f}}\right ) c \,e^{5}+8 b e \,f^{2} x \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}+2 c d \,f^{2} x \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}-10 c \,e^{2} f x \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}-16 a e \,f^{2} \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}+24 b \,e^{2} f \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}+2 c d e f \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}-30 c \,e^{3} \sqrt {e f}\, \sqrt {\left (e x +d \right ) \left (f x +e \right )}\right )}{8 \sqrt {e f}\, e \sqrt {\left (e x +d \right ) \left (f x +e \right )}\, f^{3} \sqrt {f x +e}}\) | \(834\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.29, size = 573, normalized size = 2.30 \begin {gather*} \left [-\frac {{\left (c d^{2} f^{3} x - 15 \, c e^{5} - 3 \, {\left (5 \, c f x - 4 \, b f\right )} e^{4} + 2 \, {\left (6 \, b f^{2} x + 3 \, c d f - 4 \, a f^{2}\right )} e^{3} - 2 \, {\left (2 \, b d f^{2} - {\left (3 \, c d f^{2} - 4 \, a f^{3}\right )} x\right )} e^{2} - {\left (4 \, b d f^{3} x - c d^{2} f^{2}\right )} e\right )} \sqrt {f} e^{\frac {1}{2}} \log \left (8 \, d f^{2} x e + d^{2} f^{2} + 4 \, {\left (2 \, f x e + d f + e^{2}\right )} \sqrt {f x + e} \sqrt {x e + d} \sqrt {f} e^{\frac {1}{2}} + 8 \, f x e^{3} + 2 \, {\left (4 \, f^{2} x^{2} + 3 \, d f\right )} e^{2} + e^{4}\right ) - 4 \, {\left (c d f^{3} x e - 15 \, c f e^{4} - {\left (5 \, c f^{2} x - 12 \, b f^{2}\right )} e^{3} + {\left (2 \, c f^{3} x^{2} + 4 \, b f^{3} x + c d f^{2} - 8 \, a f^{3}\right )} e^{2}\right )} \sqrt {f x + e} \sqrt {x e + d}}{16 \, {\left (f^{5} x e^{2} + f^{4} e^{3}\right )}}, \frac {{\left (c d^{2} f^{3} x - 15 \, c e^{5} - 3 \, {\left (5 \, c f x - 4 \, b f\right )} e^{4} + 2 \, {\left (6 \, b f^{2} x + 3 \, c d f - 4 \, a f^{2}\right )} e^{3} - 2 \, {\left (2 \, b d f^{2} - {\left (3 \, c d f^{2} - 4 \, a f^{3}\right )} x\right )} e^{2} - {\left (4 \, b d f^{3} x - c d^{2} f^{2}\right )} e\right )} \sqrt {-f e} \arctan \left (\frac {{\left (2 \, f x e + d f + e^{2}\right )} \sqrt {f x + e} \sqrt {-f e} \sqrt {x e + d}}{2 \, {\left (d f^{2} x e + f x e^{3} + {\left (f^{2} x^{2} + d f\right )} e^{2}\right )}}\right ) + 2 \, {\left (c d f^{3} x e - 15 \, c f e^{4} - {\left (5 \, c f^{2} x - 12 \, b f^{2}\right )} e^{3} + {\left (2 \, c f^{3} x^{2} + 4 \, b f^{3} x + c d f^{2} - 8 \, a f^{3}\right )} e^{2}\right )} \sqrt {f x + e} \sqrt {x e + d}}{8 \, {\left (f^{5} x e^{2} + f^{4} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x} \left (a + b x + c x^{2}\right )}{\left (e + f x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}\,\left (c\,x^2+b\,x+a\right )}{{\left (e+f\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________