Integrand size = 21, antiderivative size = 566 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {16 \sqrt {-a} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Time = 0.45 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {749, 847, 829, 858, 733, 435, 430} \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=-\frac {16 \sqrt {-a} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {8 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (177 a^2 e^4+113 a c d^2 e^2+32 c^2 d^4\right )-3 e x \left (-77 a^2 e^4+27 a c d^2 e^2+8 c^2 d^4\right )\right )}{9009 e^5}+\frac {16 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-231 a^3 e^6+258 a^2 c d^2 e^4+137 a c^2 d^4 e^2+32 c^3 d^6\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{9009 \sqrt {c} e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {20 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (4 d \left (5 a e^2+2 c d^2\right )-7 e x \left (c d^2-11 a e^2\right )\right )}{9009 e^3}+\frac {2 \left (a+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac {20 d \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{143 e} \]
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Rule 430
Rule 435
Rule 733
Rule 749
Rule 829
Rule 847
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {10 \int (a e-c d x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2} \, dx}{13 e} \\ & = -\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {20 \int \frac {\left (6 a c d e-\frac {1}{2} c \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{143 c e} \\ & = \frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {80 \int \frac {\left (\frac {1}{4} a c^2 d e \left (c d^2+97 a e^2\right )-\frac {1}{4} c^2 \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{3003 c^2 e^3} \\ & = \frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {64 \int \frac {a c^3 d e \left (c^2 d^4+4 a c d^2 e^2+51 a^2 e^4\right )-\frac {1}{8} c^3 \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{9009 c^3 e^5} \\ & = \frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {\left (8 d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{9009 e^6}-\frac {\left (8 \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{9009 e^6} \\ & = \frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}-\frac {\left (16 a \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\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 )}{9009 \sqrt {-a} \sqrt {c} 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 d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\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 )}{9009 \sqrt {-a} \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {8 \sqrt {d+e x} \left (d \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\right )-3 e \left (8 c^2 d^4+27 a c d^2 e^2-77 a^2 e^4\right ) x\right ) \sqrt {a+c x^2}}{9009 e^5}+\frac {20 \sqrt {d+e x} \left (4 d \left (2 c d^2+5 a e^2\right )-7 e \left (c d^2-11 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{9009 e^3}-\frac {20 d \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{143 e}+\frac {2 (d+e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 e}+\frac {16 \sqrt {-a} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\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 )}{9009 \sqrt {c} e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} d \left (c d^2+a e^2\right ) \left (32 c^2 d^4+113 a c d^2 e^2+177 a^2 e^4\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 )}{9009 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 26.38 (sec) , antiderivative size = 790, normalized size of antiderivative = 1.40 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {d+e x} \left (e^2 \left (a+c x^2\right ) \left (a^2 e^4 (971 d+2387 e x)+2 a c e^2 \left (266 d^3-197 d^2 e x+163 d e^2 x^2+1078 e^3 x^3\right )+c^2 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )-\frac {8 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^3 d^6+137 a c^2 d^4 e^2+258 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{7/2} d^7+32 \sqrt {a} c^3 d^6 e-137 i a c^{5/2} d^5 e^2+137 a^{3/2} c^2 d^4 e^3-258 i a^2 c^{3/2} d^3 e^4+258 a^{5/2} c d^2 e^5+231 i a^3 \sqrt {c} d e^6-231 a^{7/2} e^7\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^3 d^6+8 i \sqrt {a} c^{5/2} d^5 e+137 a c^2 d^4 e^2+32 i a^{3/2} c^{3/2} d^3 e^3+258 a^2 c d^2 e^4+408 i a^{5/2} \sqrt {c} d e^5-231 a^3 e^6\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{9009 e^7 \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1195\) vs. \(2(482)=964\).
Time = 2.82 (sec) , antiderivative size = 1196, normalized size of antiderivative = 2.11
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1196\) |
elliptic | \(\text {Expression too large to display}\) | \(1630\) |
default | \(\text {Expression too large to display}\) | \(2332\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.74 \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{7} + 161 \, a c^{2} d^{5} e^{2} + 354 \, a^{2} c d^{3} e^{4} + 993 \, a^{3} d e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{3} d^{6} e + 137 \, a c^{2} d^{4} e^{3} + 258 \, a^{2} c d^{2} e^{5} - 231 \, a^{3} e^{7}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (693 \, c^{3} e^{7} x^{5} + 63 \, c^{3} d e^{6} x^{4} + 128 \, c^{3} d^{5} e^{2} + 532 \, a c^{2} d^{3} e^{4} + 971 \, a^{2} c d e^{6} - 14 \, {\left (5 \, c^{3} d^{2} e^{5} - 154 \, a c^{2} e^{7}\right )} x^{3} + 2 \, {\left (40 \, c^{3} d^{3} e^{4} + 163 \, a c^{2} d e^{6}\right )} x^{2} - {\left (96 \, c^{3} d^{4} e^{3} + 394 \, a c^{2} d^{2} e^{5} - 2387 \, a^{2} c e^{7}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{27027 \, c e^{7}} \]
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\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {5}{2}} \sqrt {d + e x}\, dx \]
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\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \]
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\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d} \,d x } \]
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Timed out. \[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,\sqrt {d+e\,x} \,d x \]
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