Optimal. Leaf size=236 \[ \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {699, 705, 704,
313, 227, 1213, 435} \begin {gather*} \frac {\sqrt {d} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\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 \sqrt {a+b x+c x^2}}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\text {ArcSin}\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 \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{5 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 313
Rule 435
Rule 699
Rule 704
Rule 705
Rule 1213
Rubi steps
\begin {align*} \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{10 c}\\ &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (\left (b^2-4 a c\right ) \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}{10 c \sqrt {a+b x+c x^2}}\\ &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 d \sqrt {a+b x+c x^2}}\\ &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}+\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 {a+b x+c x^2}}\\ &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}+\frac {\left (b^2-4 a c\right )^{7/4} \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 )}{5 c^2 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 {a+b x+c x^2}}\\ &=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right )^{7/4} \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 )}{5 c^2 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{7/4} \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 )}{5 c^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 91, normalized size = 0.39 \begin {gather*} \frac {(d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{6 c d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs.
\(2(198)=396\).
time = 0.95, size = 492, normalized size = 2.08 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.65, size = 98, normalized size = 0.42 \begin {gather*} \frac {\sqrt {2} \sqrt {c^{2} d} {\left (b^{2} - 4 \, a c\right )} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (2 \, c^{2} x + b c\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{5 \, c^{2}} \end {gather*}
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 \begin {gather*} \int \sqrt {d \left (b + 2 c x\right )} \sqrt {a + b x + c x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \sqrt {b\,d+2\,c\,d\,x}\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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