Optimal. Leaf size=268 \[ \frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {684, 693, 691, 689, 221} \[ \frac {\sqrt {a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 684
Rule 689
Rule 691
Rule 693
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx}{10 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}} \, dx}{220 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{308 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{924 c^2 \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{924 c^2 \left (b^2-4 a c\right )^2 d^8 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{462 c^3 \left (b^2-4 a c\right )^2 d^9 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{462 c^3 \left (b^2-4 a c\right )^{7/4} d^{17/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 107, normalized size = 0.40 \[ \frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \sqrt {d (b+2 c x)} \, _2F_1\left (-\frac {15}{4},-\frac {3}{2};-\frac {11}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{120 c^2 d^9 (b+2 c x)^8 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{512 \, c^{9} d^{9} x^{9} + 2304 \, b c^{8} d^{9} x^{8} + 4608 \, b^{2} c^{7} d^{9} x^{7} + 5376 \, b^{3} c^{6} d^{9} x^{6} + 4032 \, b^{4} c^{5} d^{9} x^{5} + 2016 \, b^{5} c^{4} d^{9} x^{4} + 672 \, b^{6} c^{3} d^{9} x^{3} + 144 \, b^{7} c^{2} d^{9} x^{2} + 18 \, b^{8} c d^{9} x + b^{9} d^{9}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {17}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1431, normalized size = 5.34 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {17}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{17/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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