Optimal. Leaf size=325 \[ \frac {112 c^2 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}+\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}-\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}+\frac {28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}} \]
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Rubi [A] time = 0.28, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {687, 693, 691, 690, 307, 221, 1199, 424} \[ \frac {112 c^2 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}+\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}-\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}+\frac {28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 687
Rule 690
Rule 691
Rule 693
Rule 1199
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}-\frac {(14 c) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {\left (28 c^2\right ) \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (28 c^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (28 c^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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}{\left (b^2-4 a c\right )^3 d^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{\left (b^2-4 a c\right )^3 d^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}+\frac {\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}}-\frac {\left (56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}+\frac {56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}-\frac {\left (56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}+\frac {56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 97, normalized size = 0.30 \[ -\frac {32 c \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac {1}{4},\frac {5}{2};\frac {3}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{d \left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} \sqrt {d (b+2 c x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{4 \, c^{5} d^{2} x^{8} + 16 \, b c^{4} d^{2} x^{7} + {\left (25 \, b^{2} c^{3} + 12 \, a c^{4}\right )} d^{2} x^{6} + {\left (19 \, b^{3} c^{2} + 36 \, a b c^{3}\right )} d^{2} x^{5} + a^{3} b^{2} d^{2} + {\left (7 \, b^{4} c + 39 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{2} x^{4} + {\left (b^{5} + 18 \, a b^{3} c + 24 \, a^{2} b c^{2}\right )} d^{2} x^{3} + {\left (3 \, a b^{4} + 15 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{2} x^{2} + {\left (3 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{2} x}, x\right ) \]
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 \[ \int \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 877, normalized size = 2.70 \[ \frac {2 \left (-168 c^{4} x^{4}+168 \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 )-42 \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 )-336 b \,c^{3} x^{3}+168 \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 )-280 a \,c^{3} x^{2}-42 \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 )-182 b^{2} c^{2} x^{2}+168 \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^{2} c^{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 )-42 \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 )-280 a b \,c^{2} x -14 b^{3} c x -96 a^{2} c^{2}-22 a \,b^{2} c +b^{4}\right ) \sqrt {\left (2 c x +b \right ) d}}{3 \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} d^{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 \[ \int \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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