Optimal. Leaf size=485 \[ \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {c^{3/2} d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\sqrt {c} \left (3 c d^2-5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.33, antiderivative size = 485, 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 = {755, 849, 858,
733, 435, 430} \begin {gather*} -\frac {c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 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 )}{3 \sqrt {-a} \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 {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^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 \sqrt {-a} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {c d e \sqrt {a+c x^2} \left (3 c d^2-29 a e^2\right )}{3 a \sqrt {d+e x} \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (3 c d^2-5 a e^2\right )}{3 a (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^{3/2} \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 849
Rule 858
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {5 a e^2}{2}-\frac {3}{2} c d e x}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \int \frac {6 a c d e^2+\frac {1}{4} c e \left (3 c d^2-5 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {4 \int \frac {-\frac {1}{8} a c e^2 \left (27 c d^2-5 a e^2\right )+\frac {1}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (c^2 d \left (3 c d^2-29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^3}+\frac {\left (c \left (3 c d^2-5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (c^{3/2} d \left (3 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 )}{3 \sqrt {-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 (\sqrt {c} \left (3 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 )}{3 \sqrt {-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}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {c^{3/2} d \left (3 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 )}{3 \sqrt {-a} \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 {\sqrt {c} \left (3 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 )}{3 \sqrt {-a} \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 = 22.32, size = 634, normalized size = 1.31 \begin {gather*} \frac {-2 a e^3 \left (c d^2+a e^2\right ) \left (a+c x^2\right )-20 a c d e^3 (d+e x) \left (a+c x^2\right )+3 c (d+e x)^2 \left (-a^2 e^3+c^2 d^3 x+3 a c d e (d-e x)\right )+\frac {c (d+e x) \left (-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-29 a^2 e^2+3 c^2 d^2 x^2+a c \left (3 d^2-29 e^2 x^2\right )\right )+\sqrt {c} d \left (3 i c^{3/2} d^3-3 \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 (3 c^{3/2} d^3+27 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 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}}{3 a \left (c d^2+a e^2\right )^3 (d+e x)^{3/2} \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. \(2622\) vs.
\(2(409)=818\).
time = 0.47, size = 2623, normalized size = 5.41
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 e \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 \left (e^{2} a +c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{2}}-\frac {20 \left (c e \,x^{2}+a e \right ) e^{2} c d}{3 \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 (c e x +c d \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) c d x}{2 \left (e^{2} a +c \,d^{2}\right )^{3} a}+\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (-\frac {e^{2} c}{3 \left (e^{2} a +c \,d^{2}\right )^{2}}+\frac {10 d^{2} e^{2} c^{2}}{3 \left (e^{2} a +c \,d^{2}\right )^{3}}-\frac {\left (e^{2} a -c \,d^{2}\right ) c}{\left (e^{2} a +c \,d^{2}\right )^{2} a}+\frac {c \,e^{2} \left (e^{2} a -3 c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c^{2} d^{2} \left (3 e^{2} a -c \,d^{2}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3} a}\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 {10 d \,e^{3} c^{2}}{3 \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c^{2} d e \left (3 e^{2} a -c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3} a}\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}}\) | \(945\) |
default | \(\text {Expression too large to display}\) | \(2623\) |
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 = 0.38, size = 883, normalized size = 1.82 \begin {gather*} \frac {{\left (3 \, c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6} - 15 \, {\left (a^{2} c x^{4} + a^{3} x^{2}\right )} e^{6} - 30 \, {\left (a^{2} c d x^{3} + a^{3} d x\right )} e^{5} + {\left (52 \, a c^{2} d^{2} x^{4} + 37 \, a^{2} c d^{2} x^{2} - 15 \, a^{3} d^{2}\right )} e^{4} + 104 \, {\left (a c^{2} d^{3} x^{3} + a^{2} c d^{3} x\right )} e^{3} + {\left (3 \, c^{3} d^{4} x^{4} + 55 \, a c^{2} d^{4} x^{2} + 52 \, a^{2} c d^{4}\right )} e^{2} + 6 \, {\left (c^{3} d^{5} x^{3} + a 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 (29 \, {\left (a c^{2} d x^{4} + a^{2} c d x^{2}\right )} e^{5} + 58 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{4} - {\left (3 \, c^{3} d^{3} x^{4} - 26 \, a c^{2} d^{3} x^{2} - 29 \, a^{2} c d^{3}\right )} e^{3} - 6 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} - 3 \, {\left (c^{3} d^{5} x^{2} + a 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 \, {\left (3 \, c^{3} d^{5} x e - {\left (5 \, a^{2} c x^{2} + 2 \, a^{3}\right )} e^{6} - {\left (29 \, a c^{2} d x^{3} + 26 \, a^{2} c d x\right )} e^{5} - {\left (31 \, a c^{2} d^{2} x^{2} + 25 \, a^{2} c d^{2}\right )} e^{4} + 3 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}}{9 \, {\left ({\left (a^{4} c x^{4} + a^{5} x^{2}\right )} e^{9} + 2 \, {\left (a^{4} c d x^{3} + a^{5} d x\right )} e^{8} + {\left (3 \, a^{3} c^{2} d^{2} x^{4} + 4 \, a^{4} c d^{2} x^{2} + a^{5} d^{2}\right )} e^{7} + 6 \, {\left (a^{3} c^{2} d^{3} x^{3} + a^{4} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{4} + 2 \, a^{3} c^{2} d^{4} x^{2} + a^{4} c d^{4}\right )} e^{5} + 6 \, {\left (a^{2} c^{3} d^{5} x^{3} + a^{3} c^{2} d^{5} x\right )} e^{4} + {\left (a c^{4} d^{6} x^{4} + 4 \, a^{2} c^{3} d^{6} x^{2} + 3 \, a^{3} c^{2} d^{6}\right )} e^{3} + 2 \, {\left (a c^{4} d^{7} x^{3} + a^{2} c^{3} d^{7} x\right )} e^{2} + {\left (a c^{4} d^{8} x^{2} + a^{2} c^{3} d^{8}\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 {3}{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 {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\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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