Optimal. Leaf size=430 \[ \frac {8 c \sqrt {d+e x} \left (32 c d^2+5 a e^2-24 c d e x\right ) \sqrt {a+c x^2}}{21 e^5}+\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {16 \sqrt {-a} c^{3/2} d \left (32 c d^2+29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \sqrt {c} \left (c d^2+a e^2\right ) \left (32 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {747, 827, 829,
858, 733, 435, 430} \begin {gather*} \frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (29 a e^2+32 c d^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 )}{21 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {16 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (5 a e^2+32 c d^2\right ) \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 )}{21 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {8 c \sqrt {a+c x^2} \sqrt {d+e x} \left (5 a e^2+32 c d^2-24 c d e x\right )}{21 e^5}+\frac {20 c \left (a+c x^2\right )^{3/2} (8 d+e x)}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 747
Rule 827
Rule 829
Rule 858
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3 e}\\ &=\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {(20 c) \int \frac {(-a e+8 c d x) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^3}\\ &=\frac {8 c \sqrt {d+e x} \left (32 c d^2+5 a e^2-24 c d e x\right ) \sqrt {a+c x^2}}{21 e^5}+\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {16 \int \frac {-\frac {1}{2} a c e \left (8 c d^2+5 a e^2\right )+\frac {1}{2} c^2 d \left (32 c d^2+29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{21 e^5}\\ &=\frac {8 c \sqrt {d+e x} \left (32 c d^2+5 a e^2-24 c d e x\right ) \sqrt {a+c x^2}}{21 e^5}+\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {\left (8 c \left (c d^2+a e^2\right ) \left (32 c d^2+5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{21 e^6}-\frac {\left (8 c^2 d \left (32 c d^2+29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{21 e^6}\\ &=\frac {8 c \sqrt {d+e x} \left (32 c d^2+5 a e^2-24 c d e x\right ) \sqrt {a+c x^2}}{21 e^5}+\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}-\frac {\left (16 a c^{3/2} d \left (32 c d^2+29 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 )}{21 \sqrt {-a} e^6 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (16 a \sqrt {c} \left (c d^2+a e^2\right ) \left (32 c d^2+5 a e^2\right ) \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 )}{21 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {8 c \sqrt {d+e x} \left (32 c d^2+5 a e^2-24 c d e x\right ) \sqrt {a+c x^2}}{21 e^5}+\frac {20 c (8 d+e x) \left (a+c x^2\right )^{3/2}}{21 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}}+\frac {16 \sqrt {-a} c^{3/2} d \left (32 c d^2+29 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 )}{21 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \sqrt {c} \left (c d^2+a e^2\right ) \left (32 c d^2+5 a e^2\right ) \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 )}{21 e^6 \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 = 22.29, size = 637, normalized size = 1.48 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (-7 a^2 e^4+2 a c e^2 \left (50 d^2+65 d e x+8 e^2 x^2\right )+c^2 \left (128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right )\right )}{e^5 (d+e x)^2}-\frac {16 c \left (d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (29 a^2 e^2+32 c^2 d^2 x^2+a c \left (32 d^2+29 e^2 x^2\right )\right )+\sqrt {c} d \left (-32 i c^{3/2} d^3+32 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+29 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} e \left (32 c^{3/2} d^3+8 i \sqrt {a} c d^2 e+29 a \sqrt {c} d e^2+5 i 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} 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 )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{21 \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. \(2645\) vs.
\(2(352)=704\).
time = 0.66, size = 2646, normalized size = 6.15
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {28 \left (c e \,x^{2}+a e \right ) \left (e^{2} a +c \,d^{2}\right ) c d}{3 e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c^{2} x^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7 e^{3}}-\frac {8 c^{2} d x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7 e^{4}}+\frac {2 \left (\frac {3 c^{2} \left (e^{2} a +c \,d^{2}\right )}{e^{4}}-\frac {5 c^{2} a}{7 e^{2}}+\frac {16 c^{3} d^{2}}{7 e^{4}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (\frac {c \left (3 a^{2} e^{4}+9 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{e^{6}}-\frac {\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c}{3 e^{6}}-\frac {14 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )}{3 e^{6}}+\frac {8 d^{2} c^{2} a}{7 e^{4}}-\frac {\left (\frac {3 c^{2} \left (e^{2} a +c \,d^{2}\right )}{e^{4}}-\frac {5 c^{2} a}{7 e^{2}}+\frac {16 c^{3} d^{2}}{7 e^{4}}\right ) a}{3 c}\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 {2 \left (-\frac {2 c^{2} d \left (3 e^{2} a +2 c \,d^{2}\right )}{e^{5}}-\frac {14 d \left (e^{2} a +c \,d^{2}\right ) c^{2}}{3 e^{5}}+\frac {8 d \,c^{2} a}{7 e^{3}}-\frac {2 \left (\frac {3 c^{2} \left (e^{2} a +c \,d^{2}\right )}{e^{4}}-\frac {5 c^{2} a}{7 e^{2}}+\frac {16 c^{3} d^{2}}{7 e^{4}}\right ) d}{3 e}\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 )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(1041\) |
risch | \(\text {Expression too large to display}\) | \(2420\) |
default | \(\text {Expression too large to display}\) | \(2646\) |
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 = 0.62, size = 439, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (8 \, {\left (64 \, c^{2} d^{5} x e + 32 \, c^{2} d^{6} + 106 \, a c d^{3} x e^{3} + 15 \, a^{2} x^{2} e^{6} + 30 \, a^{2} d x e^{5} + {\left (53 \, a c d^{2} x^{2} + 15 \, a^{2} d^{2}\right )} e^{4} + {\left (32 \, c^{2} d^{4} x^{2} + 53 \, 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 ) + 24 \, {\left (64 \, c^{2} d^{4} x e^{2} + 32 \, c^{2} d^{5} e + 29 \, a c d x^{2} e^{5} + 58 \, a c d^{2} x e^{4} + {\left (32 \, c^{2} d^{3} x^{2} + 29 \, 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 (160 \, c^{2} d^{3} x e^{3} + 128 \, c^{2} d^{4} e^{2} + {\left (3 \, c^{2} x^{4} + 16 \, a c x^{2} - 7 \, a^{2}\right )} e^{6} - 2 \, {\left (3 \, c^{2} d x^{3} - 65 \, a c d x\right )} e^{5} + 4 \, {\left (4 \, c^{2} d^{2} x^{2} + 25 \, a c d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{63 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\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 {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {5}{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 {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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