Optimal. Leaf size=227 \[ -\frac {5 d^{7/2} \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{231 c^2 \sqrt {a+b x+c x^2}}-\frac {10 d^3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c}-\frac {2 d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \]
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Rubi [A] time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {685, 692, 691, 689, 221} \[ -\frac {5 d^{7/2} \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{231 c^2 \sqrt {a+b x+c x^2}}-\frac {10 d^3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c}-\frac {2 d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 685
Rule 689
Rule 691
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx &=\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{22 c}\\ &=-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^2 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{154 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{462 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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}{462 c \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^3 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{231 c^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \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 )}{231 c^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 155, normalized size = 0.68 \[ \frac {4 \sqrt {a+x (b+c x)} (d (b+2 c x))^{7/2} \left (7 (b+2 c x)^2 (a+x (b+c x))-10 c \left (a-\frac {b^2}{4 c}\right ) \left (\frac {\left (b^2-4 a c\right ) \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))\right )\right )}{77 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]
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 \[ \int {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} \sqrt {c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 798, normalized size = 3.52 \[ \frac {\sqrt {\left (2 c x +b \right ) d}\, \sqrt {c \,x^{2}+b x +a}\, \left (1344 c^{7} x^{7}+4704 b \,c^{6} x^{6}+1728 a \,c^{6} x^{5}+6624 b^{2} c^{5} x^{5}+4320 a b \,c^{5} x^{4}+4800 b^{3} c^{4} x^{4}-256 a^{2} c^{5} x^{3}+4448 a \,b^{2} c^{4} x^{3}+1844 b^{4} c^{3} x^{3}-384 a^{2} b \,c^{4} x^{2}+2352 a \,b^{3} c^{3} x^{2}+318 b^{5} c^{2} x^{2}-640 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x +516 a \,b^{4} c^{2} x +10 b^{6} c x -320 a^{3} b \,c^{3}+320 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, a^{3} c^{3} \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+208 a^{2} b^{3} c^{2}-240 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2} \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+10 a \,b^{5} c +60 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, a \,b^{4} c \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-5 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, b^{6} \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )\right ) d^{3}}{462 \left (2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{2}} \]
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 \[ \int {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} \sqrt {c x^{2} + b x + a}\,{d x} \]
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 \[ \int {\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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 \[ \int \left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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