Optimal. Leaf size=170 \[ -\frac {792 a^2 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac {792 a^3 x \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac {2}{9},\frac {1}{2};\frac {11}{9};-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{19747 b^2 c^3 \sqrt {a+b \left (c x^3\right )^{3/2}}} \]
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Rubi [A]
time = 0.09, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {376, 348, 285,
327, 372, 371} \begin {gather*} \frac {792 a^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 )}{19747 b^2 c^3 \sqrt {a+b \left (c x^3\right )^{3/2}}}-\frac {792 a^2 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 285
Rule 327
Rule 348
Rule 371
Rule 372
Rule 376
Rubi steps
\begin {align*} \int x^9 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx &=\text {Subst}\left (\int x^9 \sqrt {a+b c^{3/2} x^{9/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\text {Subst}\left (2 \text {Subst}\left (\int x^{19} \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 {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\text {Subst}\left (\frac {1}{49} (18 a) \text {Subst}\left (\int \frac {x^{19}}{\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 {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}-\text {Subst}\left (\frac {\left (396 a^2\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right )}{1519 b c^{3/2}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {792 a^2 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\text {Subst}\left (\frac {\left (1584 a^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b c^{3/2} x^9}} \, dx,x,\sqrt {x}\right )}{19747 b^2 c^3},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {792 a^2 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\text {Subst}\left (\frac {\left (1584 a^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 )}{19747 b^2 c^3 \sqrt {a+b c^{3/2} x^{9/2}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\frac {792 a^2 x \sqrt {a+b \left (c x^3\right )^{3/2}}}{19747 b^2 c^3}+\frac {4}{49} x^{10} \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {36 a x \left (c x^3\right )^{3/2} \sqrt {a+b \left (c x^3\right )^{3/2}}}{1519 b c^3}+\frac {792 a^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 )}{19747 b^2 c^3 \sqrt {a+b \left (c x^3\right )^{3/2}}}\\ \end {align*}
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Mathematica [F]
time = 3.64, size = 0, normalized size = 0.00 \begin {gather*} \int x^9 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x^{9} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F]
time = 154.26, size = 23, normalized size = 0.14 \begin {gather*} {\rm integral}\left (\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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{9} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^9\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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