Optimal. Leaf size=452 \[ -\frac {3 \sqrt {b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{3/2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{7/4} (b+2 c x)}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{7/4} (b+2 c x)}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]
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Rubi [A] time = 0.36, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {612, 623, 305, 220, 1196} \[ -\frac {3 \sqrt {b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{3/2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{7/4} (b+2 c x)}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{7/4} (b+2 c x)}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 612
Rule 623
Rule 1196
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^{3/4} \, dx &=\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{20 c}\\ &=\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{5 c (b+2 c x)}\\ &=\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {\left (3 \left (b^2-4 a c\right )^{3/2} \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{10 c^{3/2} (b+2 c x)}+\frac {\left (3 \left (b^2-4 a c\right )^{3/2} \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {2 \sqrt {c} x^2}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{10 c^{3/2} (b+2 c x)}\\ &=\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \sqrt {b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2}}{10 c^{3/2} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}+\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{10 \sqrt {2} c^{7/4} (b+2 c x)}-\frac {3 \left (b^2-4 a c\right )^{7/4} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{20 \sqrt {2} c^{7/4} (b+2 c x)}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 119, normalized size = 0.26 \[ \frac {(b+2 c x) (a+x (b+c x))^{3/4} \left (3 \sqrt {2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )+8 \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4}\right )}{40 c \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.48, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}\, dx \]
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 \[ \int {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}}\,{d x} \]
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 \[ \int {\left (c\,x^2+b\,x+a\right )}^{3/4} \,d x \]
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 \[ \int \left (a + b x + c x^{2}\right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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