Optimal. Leaf size=124 \[ -\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{21 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {706, 703, 227}
\begin {gather*} \frac {10 e^{7/2} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{21 d}-\frac {10 e^3 \sqrt {-c^2-2 c d x-d^2 x^2+1} \sqrt {c e+d e x}}{21 d}-\frac {2 e \sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{7 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 703
Rule 706
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^{7/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {1}{7} \left (5 e^2\right ) \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {1}{21} \left (5 e^4\right ) \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {\left (10 e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{21 d}\\ &=-\frac {10 e^3 \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d}-\frac {2 e (c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d}+\frac {10 e^{7/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{21 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.06, size = 86, normalized size = 0.69 \begin {gather*} -\frac {2 e^3 \sqrt {e (c+d x)} \left (\sqrt {1-c^2-2 c d x-d^2 x^2} \left (5+3 c^2+6 c d x+3 d^2 x^2\right )-5 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )\right )}{21 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs.
\(2(104)=208\).
time = 0.76, size = 212, normalized size = 1.71
method | result | size |
default | \(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{3} \left (-6 d^{5} x^{5}-30 c \,d^{4} x^{4}-60 c^{2} d^{3} x^{3}-60 c^{3} x^{2} d^{2}-30 c^{4} d x -4 d^{3} x^{3}-6 c^{5}-12 c \,d^{2} x^{2}+5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \EllipticF \left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-12 c^{2} d x -4 c^{3}+10 d x +10 c \right )}{21 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\) | \(212\) |
risch | \(\frac {2 \left (3 d^{2} x^{2}+6 c d x +3 c^{2}+5\right ) \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{4}}{21 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {10 \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{4}}{21 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\) | \(415\) |
elliptic | \(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \left (-\frac {2 d \,e^{3} x^{2} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{7}-\frac {4 c \,e^{3} x \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{7}-\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}{3 d^{3} e}+\frac {2 \left (e^{4} c^{4}+\frac {4 c \,e^{3} \left (-c^{3} e +c e \right )}{7}+\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) \left (-\frac {3}{2} c^{2} d e +\frac {1}{2} d e \right )}{3 d^{3} e}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}+\frac {2 \left (4 c^{3} d \,e^{4}+\frac {2 d \,e^{3} \left (-2 c^{3} e +2 c e \right )}{7}+\frac {4 c \,e^{3} \left (-\frac {9}{2} c^{2} d e +\frac {3}{2} d e \right )}{7}-\frac {2 \left (\frac {18 c^{2} d^{2} e^{4}}{7}+\frac {2 d \,e^{3} \left (-\frac {15}{2} c^{2} d e +\frac {5}{2} d e \right )}{7}\right ) c}{d}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c \EllipticF \left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d x e +c e}}\right )}{\left (d x +c \right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e}\) | \(923\) |
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 = 0.38, size = 94, normalized size = 0.76 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, d^{4} x^{2} + 6 \, c d^{3} x + {\left (3 \, c^{2} + 5\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d x + c} e^{\frac {7}{2}} + 5 \, \sqrt {-d^{3} e} e^{3} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{21 \, d^{3}} \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 (e \left (c + d x\right )\right )^{\frac {7}{2}}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, 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 (c\,e+d\,e\,x\right )}^{7/2}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________