Optimal. Leaf size=137 \[ \frac {4 b^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{a} \sqrt {a x+b x^3}}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}-\frac {4 b \sqrt {a x+b x^3}}{7 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2020, 2011, 329, 220} \[ \frac {4 b^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{a} \sqrt {a x+b x^3}}-\frac {4 b \sqrt {a x+b x^3}}{7 x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 329
Rule 2011
Rule 2020
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3\right )^{3/2}}{x^6} \, dx &=-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}+\frac {1}{7} (6 b) \int \frac {\sqrt {a x+b x^3}}{x^3} \, dx\\ &=-\frac {4 b \sqrt {a x+b x^3}}{7 x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}+\frac {1}{7} \left (4 b^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx\\ &=-\frac {4 b \sqrt {a x+b x^3}}{7 x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}+\frac {\left (4 b^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{7 \sqrt {a x+b x^3}}\\ &=-\frac {4 b \sqrt {a x+b x^3}}{7 x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}+\frac {\left (8 b^2 \sqrt {x} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{7 \sqrt {a x+b x^3}}\\ &=-\frac {4 b \sqrt {a x+b x^3}}{7 x^2}-\frac {2 \left (a x+b x^3\right )^{3/2}}{7 x^5}+\frac {4 b^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{a} \sqrt {a x+b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 54, normalized size = 0.39 \[ -\frac {2 a \sqrt {x \left (a+b x^2\right )} \, _2F_1\left (-\frac {7}{4},-\frac {3}{2};-\frac {3}{4};-\frac {b x^2}{a}\right )}{7 x^4 \sqrt {\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a x} {\left (b x^{2} + a\right )}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 142, normalized size = 1.04 \[ \frac {4 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, b \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 \sqrt {b \,x^{3}+a x}}-\frac {6 \sqrt {b \,x^{3}+a x}\, b}{7 x^{2}}-\frac {2 \sqrt {b \,x^{3}+a x}\, a}{7 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^3+a\,x\right )}^{3/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________