Optimal. Leaf size=340 \[ -\frac {d (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {(d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \]
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Rubi [A]
time = 0.48, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1403, 1804,
1362, 440, 1399, 524} \begin {gather*} \frac {(d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 524
Rule 1362
Rule 1399
Rule 1403
Rule 1804
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx &=\frac {\text {Subst}\left (\int \frac {-d+x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {d}{\sqrt {a+b x^3+c x^6}}+\frac {x}{\sqrt {a+b x^3+c x^6}}\right ) \, dx,x,d+e x\right )}{e^2}\\ &=\frac {\text {Subst}\left (\int \frac {x}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3+c x^6}} \, dx,x,d+e x\right )}{e^2}\\ &=\frac {\left (\sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \, dx,x,d+e x\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {\left (d \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \, dx,x,d+e x\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\\ &=-\frac {d (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {(d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{2 e^2 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\\ \end {align*}
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Mathematica [F]
time = 10.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {a +b \left (e x +d \right )^{3}+c \left (e x +d \right )^{6}}}\, dx\]
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 \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 = 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, 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 = -0.00 \begin {gather*} \int \frac {x}{\sqrt {a+b\,{\left (d+e\,x\right )}^3+c\,{\left (d+e\,x\right )}^6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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