Optimal. Leaf size=97 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0548172, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {369, 266, 47, 51, 63, 208} \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 369
Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^4} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x^{3/2}}}{x^4} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x}}{x^3} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3}+\operatorname{Subst}\left (\frac{1}{6} \left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b \sqrt{c} x}} \, dx,x,x^{3/2}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\operatorname{Subst}\left (\frac{\left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,x^{3/2}\right )}{12 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\operatorname{Subst}\left (\frac{\left (b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b \sqrt{c}}+\frac{x^2}{b \sqrt{c}}} \, dx,x,\sqrt{a+b \sqrt{c} x^{3/2}}\right )}{6 a},\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}+\frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}\\ \end{align*}
Mathematica [F] time = 0.0414057, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^4} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.183, size = 81, normalized size = 0.8 \begin{align*} -{\frac{1}{6\,{x}^{3}} \left ( -{b}^{2}{\it Artanh} \left ({\sqrt{a+b\sqrt{c{x}^{3}}}{\frac{1}{\sqrt{a}}}} \right ){x}^{3}ca+\sqrt{c{x}^{3}}b\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{{\frac{3}{2}}}+2\,\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{5/2} \right ){a}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21687, size = 127, normalized size = 1.31 \begin{align*} -\frac{1}{6} \, b^{2} c^{\frac{3}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{\sqrt{c x} b c x + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a c} + \frac{\sqrt{\sqrt{c x} b c x + a c} a c +{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}}}{a b^{2} c^{4} x^{3}}\right )}{\left | c \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]