Optimal. Leaf size=449 \[ \frac {27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{640 b d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {27 \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{320 b d^2}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b} \]
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Rubi [A] time = 0.34, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 63, 225} \[ \frac {27\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{8/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{640 b d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {27 \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^2}{320 b d^2}+\frac {3 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 225
Rubi steps
\begin {align*} \int (a+b x)^{3/2} \sqrt [6]{c+d x} \, dx &=\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b}+\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{(c+d x)^{5/6}} \, dx}{16 b}\\ &=\frac {3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b}-\frac {\left (9 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{5/6}} \, dx}{160 b d}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{320 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b}+\frac {\left (27 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/6}} \, dx}{640 b d^2}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{320 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b}+\frac {\left (81 (b c-a d)^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 )}{320 b d^3}\\ &=-\frac {27 (b c-a d)^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{320 b d^2}+\frac {3 (b c-a d) (a+b x)^{3/2} \sqrt [6]{c+d x}}{80 b d}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x}}{8 b}+\frac {27\ 3^{3/4} (b c-a d)^{8/3} \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 )}{640 b d^3 \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 = 0.16 \[ \frac {2 (a+b x)^{5/2} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {1}{6},\frac {5}{2};\frac {7}{2};\frac {d (a+b x)}{a d-b c}\right )}{5 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{6}}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {1}{6}}\, dx \]
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 \[ \int {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{6}}\,{d x} \]
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 \[ \int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/6} \,d x \]
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 \[ \int \left (a + b x\right )^{\frac {3}{2}} \sqrt [6]{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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