Optimal. Leaf size=139 \[ \frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{3/2} \sqrt {c d f-a e g}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {894, 888, 211}
\begin {gather*} \frac {2 e \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} \sqrt {c d f-a e g}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 888
Rule 894
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {\left (2 \left (\frac {1}{2} c d e^2 f-\frac {1}{2} c d^2 e g\right )\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e g}\\ &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {\left (2 e^2 (e f-d g)\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{g}\\ &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{3/2} \sqrt {c d f-a e g}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 140, normalized size = 1.01 \begin {gather*} \frac {2 \sqrt {d+e x} \left (e \sqrt {g} \sqrt {c d f-a e g} (a e+c d x)+c d (-e f+d g) \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{c d g^{3/2} \sqrt {c d f-a e g} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 153, normalized size = 1.10
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c \,d^{2} g -\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d e f -e \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\right )}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, c d g \sqrt {\left (a e g -c d f \right ) g}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.89, size = 520, normalized size = 3.74 \begin {gather*} \left [\frac {{\left (c d^{3} g - c d f x e^{2} + {\left (c d^{2} g x - c d^{2} f\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e + 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, {\left (c d f g e - a g^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c^{2} d^{3} f g^{2} - a c d g^{3} x e^{2} + {\left (c^{2} d^{2} f g^{2} x - a c d^{2} g^{3}\right )} e}, -\frac {2 \, {\left ({\left (c d^{3} g - c d f x e^{2} + {\left (c d^{2} g x - c d^{2} f\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) - {\left (c d f g e - a g^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}\right )}}{c^{2} d^{3} f g^{2} - a c d g^{3} x e^{2} + {\left (c^{2} d^{2} f g^{2} x - a c d^{2} g^{3}\right )} e}\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 {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}}{\left (f+g\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________