Optimal. Leaf size=201 \[ \frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \]
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Rubi [A] time = 0.17, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {90, 80, 63, 240, 212, 208, 205} \begin {gather*} \frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 90
Rule 205
Rule 208
Rule 212
Rule 240
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\int \frac {-a c-\frac {1}{4} (5 b c+7 a d) x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 b d}\\ &=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 b^2 d^2}\\ &=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{c-\frac {a d}{b}+\frac {d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{8 b^3 d^2}\\ &=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^4}{b}} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{8 b^3 d^2}\\ &=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 b^{5/2} d^2}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 b^{5/2} d^2}\\ &=-\frac {(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac {x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}\\ \end {align*}
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Mathematica [C] time = 0.13, size = 123, normalized size = 0.61 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {d (a+b x)}{a d-b c}\right )-b (c+d x) (7 a d+5 b c-4 b d x)\right )}{8 b^3 d^2 \sqrt [4]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 269, normalized size = 1.34 \begin {gather*} \frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac {\sqrt [4]{a+b x} \left (\frac {7 a^2 d^3 (a+b x)}{c+d x}-11 a^2 b d^2-\frac {9 b^2 c^2 d (a+b x)}{c+d x}+6 a b^2 c d+\frac {2 a b c d^2 (a+b x)}{c+d x}+5 b^3 c^2\right )}{8 b^2 d^2 \sqrt [4]{c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.40, size = 1513, normalized size = 7.53
result too large to display
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*} \int \frac {x^{2}}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}\, dx \end {gather*}
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*} \int \frac {x^{2}}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \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 = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+b\,x\right )}^{3/4}\,{\left (c+d\,x\right )}^{1/4}} \,d x \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 \frac {x^{2}}{\left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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