Optimal. Leaf size=283 \[ \frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c d^3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}} \]
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Rubi [A] time = 0.25, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {684, 693, 691, 690, 307, 221, 1199, 424} \[ \frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c d^3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 684
Rule 690
Rule 691
Rule 693
Rule 1199
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{10 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}-\frac {\int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{10 c \left (b^2-4 a c\right ) d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {\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}{10 c \left (b^2-4 a c\right ) d^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 \left (b^2-4 a c\right ) d^5 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}+\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 )}{5 c^2 \sqrt {b^2-4 a c} d^4 \sqrt {a+b x+c x^2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 \sqrt {b^2-4 a c} d^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}+\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 )}{5 c^2 \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {a+b x+c x^2}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 \sqrt {b^2-4 a c} d^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{5 c d (b d+2 c d x)^{5/2}}+\frac {2 \sqrt {a+b x+c x^2}}{5 c \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}}-\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{5 c^2 \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {a+b x+c x^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 )}{5 c^2 \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 91, normalized size = 0.32 \[ -\frac {\sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {5}{4},-\frac {1}{2};-\frac {1}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{10 c d \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 874, normalized size = 3.09 \[ \frac {\left (-16 c^{4} x^{4}+16 \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}}}}\, a \,c^{3} x^{2} \EllipticE \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 )-4 \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}}}}\, b^{2} c^{2} x^{2} \EllipticE \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 )-32 b \,c^{3} x^{3}+16 \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}}}}\, a b \,c^{2} x \EllipticE \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 )-24 a \,c^{3} x^{2}-4 \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}}}}\, b^{3} c x \EllipticE \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 )-18 b^{2} c^{2} x^{2}+4 \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}}}}\, a \,b^{2} c \EllipticE \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 )-24 a b \,c^{2} x -\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}}}}\, b^{4} \EllipticE \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 )-2 b^{3} c x -8 a^{2} c^{2}-2 a \,b^{2} c \right ) \sqrt {\left (2 c x +b \right ) d}}{10 \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{3} c^{2} d^{4}} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x + c x^{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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