Optimal. Leaf size=444 \[ -\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} c^{3/2} \left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {747, 849, 858,
733, 435, 430} \begin {gather*} -\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {4 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (c d^2-3 a e^2\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {4 c \sqrt {a+c x^2} \left (c d^2-3 a e^2\right )}{15 e \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {4 c d \sqrt {a+c x^2}}{15 e (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 747
Rule 849
Rule 858
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {(2 c) \int \frac {x}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx}{5 e}\\ &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {(4 c) \int \frac {-\frac {3 a e}{2}-\frac {c d x}{2}}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{15 e \left (c d^2+a e^2\right )}\\ &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {(8 c) \int \frac {a c d e-\frac {1}{4} c \left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e \left (c d^2+a e^2\right )^2}\\ &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (2 c^2 \left (c d^2-3 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{15 e^2 \left (c d^2+a e^2\right )^2}+\frac {\left (2 c^2 d\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^2 \left (c d^2+a e^2\right )}\\ &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (4 a c^{3/2} \left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{15 \sqrt {-a} e^2 \left (c d^2+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a c^{3/2} d \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{15 \sqrt {-a} e^2 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {a+c x^2}}{5 e (d+e x)^{5/2}}+\frac {4 c d \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (c d^2-3 a e^2\right ) \sqrt {a+c x^2}}{15 e \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {4 \sqrt {-a} c^{3/2} \left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} c^{3/2} d \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^2 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 21.39, size = 602, normalized size = 1.36 \begin {gather*} \frac {2 \left (-e^2 \left (a+c x^2\right ) \left (3 a^2 e^4-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )+2 a c e^2 \left (5 d^2+5 d e x+3 e^2 x^2\right )\right )+\frac {2 c (d+e x)^2 \left (-e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-3 a^2 e^2+c^2 d^2 x^2+a c \left (d^2-3 e^2 x^2\right )\right )+\sqrt {c} \left (i c^{3/2} d^3-\sqrt {a} c d^2 e-3 i a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \sqrt {c} e \left (c d^2+4 i \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e^3 \left (c d^2+a e^2\right )^2 (d+e x)^{5/2} \sqrt {a+c x^2}} \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. \(3408\) vs.
\(2(366)=732\).
time = 0.48, size = 3409, normalized size = 7.68
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{4} \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{3} \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 \left (c e \,x^{2}+a e \right ) c \left (3 e^{2} a -c \,d^{2}\right )}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {2 d \,c^{2}}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )}+\frac {2 d \,c^{2} \left (3 e^{2} a -c \,d^{2}\right )}{15 e^{2} \left (e^{2} a +c \,d^{2}\right )^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {4 c^{2} \left (3 e^{2} a -c \,d^{2}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{15 e \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(793\) |
default | \(\text {Expression too large to display}\) | \(3409\) |
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 = 0.42, size = 538, normalized size = 1.21 \begin {gather*} \frac {2 \, {\left (2 \, {\left (3 \, c^{2} d^{5} x e + c^{2} d^{6} + 9 \, a c d x^{3} e^{5} + 27 \, a c d^{2} x^{2} e^{4} + {\left (c^{2} d^{3} x^{3} + 27 \, a c d^{3} x\right )} e^{3} + 3 \, {\left (c^{2} d^{4} x^{2} + 3 \, a c d^{4}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 6 \, {\left (3 \, c^{2} d^{4} x e^{2} + c^{2} d^{5} e - 3 \, a c x^{3} e^{6} - 9 \, a c d x^{2} e^{5} + {\left (c^{2} d^{2} x^{3} - 9 \, a c d^{2} x\right )} e^{4} + 3 \, {\left (c^{2} d^{3} x^{2} - a c d^{3}\right )} e^{3}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (6 \, c^{2} d^{3} x e^{3} + c^{2} d^{4} e^{2} - 10 \, a c d x e^{5} - 3 \, {\left (2 \, a c x^{2} + a^{2}\right )} e^{6} + 2 \, {\left (c^{2} d^{2} x^{2} - 5 \, a c d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{45 \, {\left (3 \, c^{2} d^{6} x e^{4} + c^{2} d^{7} e^{3} + a^{2} x^{3} e^{10} + 3 \, a^{2} d x^{2} e^{9} + {\left (2 \, a c d^{2} x^{3} + 3 \, a^{2} d^{2} x\right )} e^{8} + {\left (6 \, a c d^{3} x^{2} + a^{2} d^{3}\right )} e^{7} + {\left (c^{2} d^{4} x^{3} + 6 \, a c d^{4} x\right )} e^{6} + {\left (3 \, c^{2} d^{5} x^{2} + 2 \, a c d^{5}\right )} e^{5}\right )}} \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 \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{\frac {7}{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 \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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