Optimal. Leaf size=532 \[ \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 \sqrt {c} d \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 532, 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 = {755, 837, 849,
858, 733, 435, 430} \begin {gather*} \frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\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 )}{6 (-a)^{3/2} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {e \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (3 a e^2+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 )}{3 (-a)^{3/2} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 733
Rule 755
Rule 837
Rule 849
Rule 858
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (-4 c d^2-7 a e^2\right )-\frac {5}{2} c d e x}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {\int \frac {-\frac {3}{4} a c e^2 \left (c d^2-7 a e^2\right )+c^2 d e \left (c d^2+3 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \int \frac {-\frac {1}{8} a c^2 d e^2 \left (c d^2+33 a e^2\right )+\frac {1}{8} c^2 e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (c d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{12 a^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{6 \sqrt {-a} a \left (c d^2+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 \sqrt {c} d \left (c d^2+3 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 )}{3 \sqrt {-a} a \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{6 (-a)^{3/2} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 \sqrt {c} d \left (c d^2+3 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 )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \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 = 22.72, size = 669, normalized size = 1.26 \begin {gather*} \frac {21 a^3 e^5-4 c^3 d^4 e x^2-12 a^2 e^5 \left (a+c x^2\right )+3 a^2 c e^3 \left (-5 d^2+7 e^2 x^2\right )-a c^2 d^2 e \left (4 d^2+15 e^2 x^2\right )+\frac {2 a c \left (c d^2+a e^2\right ) (d+e x) \left (c d^2 x+a e (2 d-e x)\right )}{a+c x^2}+c (d+e x) \left (4 c^2 d^4 x+3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)\right )-\frac {i c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{e}+\frac {\sqrt {a} \sqrt {c} \left (4 c^2 d^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4\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 )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x} \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. \(3321\) vs.
\(2(455)=910\).
time = 0.48, size = 3322, normalized size = 6.24
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a c}+\frac {2 d e}{3 \left (e^{2} a +c \,d^{2}\right )^{2} c}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right ) x}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}-\frac {d e \left (21 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}-\frac {2 \left (c e \,x^{2}+a e \right ) e^{4}}{\left (e^{2} a +c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {2 c d \left (3 e^{2} a +c \,d^{2}\right )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a^{2}}-\frac {c \,e^{2} d \left (21 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a}+\frac {c d \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{6 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}+\frac {e^{4} c d}{\left (e^{2} a +c \,d^{2}\right )^{3}}\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 {c e \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}+\frac {e^{5} c}{\left (e^{2} a +c \,d^{2}\right )^{3}}\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}}\) | \(1000\) |
default | \(\text {Expression too large to display}\) | \(3322\) |
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 = 1.92, size = 1103, normalized size = 2.07 \begin {gather*} \frac {2 \, {\left (2 \, c^{4} d^{6} x^{4} + 4 \, a c^{3} d^{6} x^{2} + 2 \, a^{2} c^{2} d^{6} + 39 \, {\left (a^{2} c^{2} d x^{5} + 2 \, a^{3} c d x^{3} + a^{4} d x\right )} e^{5} + 39 \, {\left (a^{2} c^{2} d^{2} x^{4} + 2 \, a^{3} c d^{2} x^{2} + a^{4} d^{2}\right )} e^{4} + 9 \, {\left (a c^{3} d^{3} x^{5} + 2 \, a^{2} c^{2} d^{3} x^{3} + a^{3} c d^{3} x\right )} e^{3} + 9 \, {\left (a c^{3} d^{4} x^{4} + 2 \, a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{2} + 2 \, {\left (c^{4} d^{5} x^{5} + 2 \, a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e\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 ) - 3 \, {\left (21 \, {\left (a^{2} c^{2} x^{5} + 2 \, a^{3} c x^{3} + a^{4} x\right )} e^{6} + 21 \, {\left (a^{2} c^{2} d x^{4} + 2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{5} - 15 \, {\left (a c^{3} d^{2} x^{5} + 2 \, a^{2} c^{2} d^{2} x^{3} + a^{3} c d^{2} x\right )} e^{4} - 15 \, {\left (a c^{3} d^{3} x^{4} + 2 \, a^{2} c^{2} d^{3} x^{2} + a^{3} c d^{3}\right )} e^{3} - 4 \, {\left (c^{4} d^{4} x^{5} + 2 \, a c^{3} d^{4} x^{3} + a^{2} c^{2} d^{4} x\right )} e^{2} - 4 \, {\left (c^{4} d^{5} x^{4} + 2 \, a c^{3} d^{5} x^{2} + a^{2} c^{2} d^{5}\right )} e\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 \, \sqrt {c x^{2} + a} {\left ({\left (21 \, a^{2} c^{2} x^{4} + 35 \, a^{3} c x^{2} + 12 \, a^{4}\right )} e^{6} - 2 \, {\left (6 \, a^{2} c^{2} d x^{3} + 7 \, a^{3} c d x\right )} e^{5} - {\left (15 \, a c^{3} d^{2} x^{4} + 36 \, a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{4} - 4 \, {\left (4 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{3} - {\left (4 \, c^{4} d^{4} x^{4} + 7 \, a c^{3} d^{4} x^{2} + 5 \, a^{2} c^{2} d^{4}\right )} e^{2} - 2 \, {\left (2 \, c^{4} d^{5} x^{3} + 3 \, a c^{3} d^{5} x\right )} e\right )} \sqrt {x e + d}}{18 \, {\left ({\left (a^{5} c^{2} x^{5} + 2 \, a^{6} c x^{3} + a^{7} x\right )} e^{8} + {\left (a^{5} c^{2} d x^{4} + 2 \, a^{6} c d x^{2} + a^{7} d\right )} e^{7} + 3 \, {\left (a^{4} c^{3} d^{2} x^{5} + 2 \, a^{5} c^{2} d^{2} x^{3} + a^{6} c d^{2} x\right )} e^{6} + 3 \, {\left (a^{4} c^{3} d^{3} x^{4} + 2 \, a^{5} c^{2} d^{3} x^{2} + a^{6} c d^{3}\right )} e^{5} + 3 \, {\left (a^{3} c^{4} d^{4} x^{5} + 2 \, a^{4} c^{3} d^{4} x^{3} + a^{5} c^{2} d^{4} x\right )} e^{4} + 3 \, {\left (a^{3} c^{4} d^{5} x^{4} + 2 \, a^{4} c^{3} d^{5} x^{2} + a^{5} c^{2} d^{5}\right )} e^{3} + {\left (a^{2} c^{5} d^{6} x^{5} + 2 \, a^{3} c^{4} d^{6} x^{3} + a^{4} c^{3} d^{6} x\right )} e^{2} + {\left (a^{2} c^{5} d^{7} x^{4} + 2 \, a^{3} c^{4} d^{7} x^{2} + a^{4} c^{3} d^{7}\right )} e\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 {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{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 {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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