Optimal. Leaf size=165 \[ -\frac {b \sqrt {d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{8 c^{7/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt {\frac {d}{x}}\right )}{12 c^3}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x} \]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1970, 1357, 742, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (16 a c-15 b^2 d+10 b c \sqrt {\frac {d}{x}}\right )}{12 c^3}-\frac {b \sqrt {d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{8 c^{7/2}}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 742
Rule 779
Rule 1357
Rule 1970
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )}{d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^2}\\ &=-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x}-\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-2 a-\frac {5 b x}{2}\right )}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{3 c d}\\ &=\frac {\left (16 a c-5 b \left (3 b d-2 c \sqrt {\frac {d}{x}}\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 c^3}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x}-\frac {\left (b \left (12 a c-5 b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{8 c^3}\\ &=\frac {\left (16 a c-5 b \left (3 b d-2 c \sqrt {\frac {d}{x}}\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 c^3}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x}-\frac {\left (b \left (12 a c-5 b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {4 c}{d}-x^2} \, dx,x,\frac {b+\frac {2 c \sqrt {\frac {d}{x}}}{d}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{4 c^3}\\ &=\frac {\left (16 a c-5 b \left (3 b d-2 c \sqrt {\frac {d}{x}}\right )\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{12 c^3}-\frac {2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 c x}-\frac {b \sqrt {d} \left (12 a c-5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \left (b+\frac {2 c \sqrt {\frac {d}{x}}}{d}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{8 c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.06, size = 273, normalized size = 1.65 \begin {gather*} \frac {\left (5 b^3 d^{3/2}-12 a b c \sqrt {d}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \left (2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}-2 \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}}\right )}{b \sqrt {d}}\right )}{8 c^{7/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (16 a c d-15 b^2 d^2+10 b c d \sqrt {\frac {d}{x}}-\frac {8 c^2 d}{x}\right )}{12 c^3 d}+\frac {\sqrt {\frac {c}{d}} \left (12 a b c d-5 b^3 d^2\right ) \log \left (-8 c d \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+4 a c d-b^2 d^2+4 b c d \sqrt {\frac {d}{x}}+\frac {8 c^2 d}{x}\right )}{16 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] 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]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 267, normalized size = 1.62 \begin {gather*} -\frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-15 \left (\frac {d}{x}\right )^{\frac {3}{2}} b^{3} c \,x^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+36 \sqrt {\frac {d}{x}}\, a b \,c^{2} x^{2} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+30 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, b^{2} c^{\frac {3}{2}} d x -32 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,c^{\frac {5}{2}} x -20 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, b \,c^{\frac {5}{2}} x +16 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {7}{2}}\right )}{24 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{\frac {9}{2}} x} \end {gather*}
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*} \int \frac {1}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{3}}\,{d x} \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 {1}{x^3\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \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 {1}{x^{3} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________