Optimal. Leaf size=139 \[ -\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^3 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{3/2}}-\frac {45 b^2 c^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.01, 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 = {376, 348, 283,
331, 372, 371} \begin {gather*} -\frac {45 b^2 c^3 x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a}+1} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 283
Rule 331
Rule 348
Rule 371
Rule 372
Rule 376
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x^9} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\text {Subst}\left (2 \text {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^9}}{x^{17}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}+\text {Subst}\left (\frac {1}{16} \left (9 b c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\text {Subst}\left (\frac {\left (45 b^2 c^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right )}{224 a},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\text {Subst}\left (\frac {\left (45 b^2 c^3 \sqrt {1+\frac {b c^{3/2} x^{9/2}}{a}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+\frac {b c^{3/2} x^9}{a}}} \, dx,x,\sqrt {x}\right )}{224 a \sqrt {a+b c^{3/2} x^{9/2}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{8 x^8}-\frac {9 b c^5 x^7 \sqrt {a+b \left (c x^3\right )^{3/2}}}{112 a \left (c x^3\right )^{7/2}}-\frac {45 b^2 c^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{448 a \sqrt {a+b \left (c x^3\right )^{3/2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 3.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^9} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{x^{9}}\, dx\]
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 [F]
time = 152.58, size = 23, normalized size = 0.17 \begin {gather*} {\rm integral}\left (\frac {\sqrt {\sqrt {c x^{3}} b c x^{3} + a}}{x^{9}}, x\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 {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________