Optimal. Leaf size=281 \[ -\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{9 c d} \]
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Rubi [A] time = 0.25, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {685, 691, 690, 307, 221, 1199, 424} \[ -\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{30 c^2 d}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\sqrt {d} \left (b^2-4 a c\right )^{11/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{9 c d} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 424
Rule 685
Rule 690
Rule 691
Rule 1199
Rubi steps
\begin {align*} \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (b^2-4 a c\right ) \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx}{6 c}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{60 c^2}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (\left (b^2-4 a c\right )^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}{60 c^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (\left (b^2-4 a c\right )^2 \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 )}{30 c^3 d \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (\left (b^2-4 a c\right )^{5/2} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{5/2} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{5/2} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{30 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c d}+\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{11/4} \sqrt {d} \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 )}{30 c^3 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 99, normalized size = 0.35 \[ -\frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} (d (b+2 c x))^{3/2} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{24 c^2 d \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 700, normalized size = 2.49 \[ \frac {\sqrt {\left (2 c x +b \right ) d}\, \sqrt {c \,x^{2}+b x +a}\, \left (80 c^{6} x^{6}+240 b \,c^{5} x^{5}+256 a \,c^{5} x^{4}+236 b^{2} c^{4} x^{4}+512 a b \,c^{4} x^{3}+72 b^{3} c^{3} x^{3}+176 a^{2} c^{4} x^{2}+296 a \,b^{2} c^{3} x^{2}-10 b^{4} c^{2} x^{2}+192 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a^{3} c^{3} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-144 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a^{2} b^{2} c^{2} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+176 a^{2} b \,c^{3} x +36 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a \,b^{4} c \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+40 a \,b^{3} c^{2} x -3 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, b^{6} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-6 b^{5} c x +44 a^{2} b^{2} c^{2}-6 a \,b^{4} c \right )}{180 \left (2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{3}} \]
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 \[ \int \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \sqrt {b\,d+2\,c\,d\,x}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.91, size = 264, normalized size = 0.94 \[ \frac {a \left (b d + 2 c d x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{4 c d \Gamma \left (\frac {7}{4}\right )} - \frac {b^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{16 c^{2} d \Gamma \left (\frac {7}{4}\right )} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{16 c^{2} d^{3} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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