Optimal. Leaf size=890 \[ \frac {243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {243 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {81 \sqrt {a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac {9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]
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Rubi [A] time = 0.95, antiderivative size = 890, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {50, 63, 308, 225, 1881} \[ \frac {243 \sqrt [4]{3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) (b c-a d)^{10/3}}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {243 \left (1+\sqrt {3}\right ) \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {81 \sqrt {a+b x} (c+d x)^{5/6} (b c-a d)^2}{224 d^3}-\frac {9 (a+b x)^{3/2} (c+d x)^{5/6} (b c-a d)}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 225
Rule 308
Rule 1881
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{\sqrt [6]{c+d x}} \, dx &=\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {(3 (b c-a d)) \int \frac {(a+b x)^{3/2}}{\sqrt [6]{c+d x}} \, dx}{4 d}\\ &=-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {\left (27 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt [6]{c+d x}} \, dx}{56 d^2}\\ &=\frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {\left (81 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}} \, dx}{448 d^3}\\ &=\frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}-\frac {\left (243 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{224 d^4}\\ &=\frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {\left (243 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) (b c-a d)^{2/3}-2 b^{2/3} x^4}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4}+\frac {\left (243 \left (1-\sqrt {3}\right ) (b c-a d)^{11/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{448 b^{2/3} d^4}\\ &=\frac {81 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{5/6}}{224 d^3}-\frac {9 (b c-a d) (a+b x)^{3/2} (c+d x)^{5/6}}{28 d^2}+\frac {3 (a+b x)^{5/2} (c+d x)^{5/6}}{10 d}+\frac {243 \left (1+\sqrt {3}\right ) (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{448 b^{2/3} d^3 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {243 \sqrt [4]{3} (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81\ 3^{3/4} \left (1-\sqrt {3}\right ) (b c-a d)^{10/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{2/3} d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 73, normalized size = 0.08 \[ \frac {2 (a+b x)^{7/2} \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{6},\frac {7}{2};\frac {9}{2};\frac {d (a+b x)}{a d-b c}\right )}{7 b \sqrt [6]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {1}{6}}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (d x +c \right )^{\frac {1}{6}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )}^{\frac {1}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{1/6}} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\sqrt [6]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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