Optimal. Leaf size=174 \[ \frac {20}{21} \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {10 \left (b^2-4 a c\right )^{9/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{21 c \sqrt {a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {706, 705, 703,
227} \begin {gather*} \frac {10 d^{7/2} \left (b^2-4 a c\right )^{9/4} \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 )}{21 c \sqrt {a+b x+c x^2}}+\frac {20}{21} d^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 703
Rule 705
Rule 706
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{7} \left (5 \left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=\frac {20}{21} \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {1}{21} \left (5 \left (b^2-4 a c\right )^2 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx\\ &=\frac {20}{21} \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (5 \left (b^2-4 a c\right )^2 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}{21 \sqrt {a+b x+c x^2}}\\ &=\frac {20}{21} \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {\left (10 \left (b^2-4 a c\right )^2 d^3 \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 )}{21 c \sqrt {a+b x+c x^2}}\\ &=\frac {20}{21} \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {4}{7} d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}+\frac {10 \left (b^2-4 a c\right )^{9/4} 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 )}{21 c \sqrt {a+b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.14, size = 166, normalized size = 0.95 \begin {gather*} \frac {2 d^3 \sqrt {d (b+2 c x)} \left (8 c \left (-5 a^2 c+2 a \left (b^2-b c x-c^2 x^2\right )+x \left (2 b^3+5 b^2 c x+6 b c^2 x^2+3 c^3 x^3\right )\right )+5 \left (b^2-4 a c\right )^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{21 c \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs.
\(2(146)=292\).
time = 0.77, size = 567, normalized size = 3.26 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.46, size = 131, normalized size = 0.75 \begin {gather*} \frac {5 \, \sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 16 \, {\left (3 \, c^{4} d^{3} x^{2} + 3 \, b c^{3} d^{3} x + {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{21 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________