Optimal. Leaf size=195 \[ -\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {745, 735, 739,
212} \begin {gather*} -\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac {3 a c^2 d \sqrt {a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 735
Rule 739
Rule 745
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {(c d) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{c d^2+a e^2}\\ &=-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {\left (3 a c^2 d\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{4 \left (c d^2+a e^2\right )^2}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}+\frac {\left (3 a^2 c^3 d\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {\left (3 a^2 c^3 d\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac {3 a c^2 d (a e-c d x) \sqrt {a+c x^2}}{8 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {c d (a e-c d x) \left (a+c x^2\right )^{3/2}}{4 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{5 \left (c d^2+a e^2\right ) (d+e x)^5}-\frac {3 a^2 c^3 d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{8 \left (c d^2+a e^2\right )^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.70, size = 253, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-8 a^4 e^5+2 c^4 d^4 x^3 (5 d+e x)-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )+a c^3 d^2 x \left (25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right )-a^2 c^2 e \left (33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right )\right )}{40 \left (c d^2+a e^2\right )^3 (d+e x)^5}+\frac {3 a^2 c^3 d \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{4 \left (-c d^2-a e^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4107\) vs.
\(2(175)=350\).
time = 0.45, size = 4108, normalized size = 21.07
method | result | size |
default | \(\text {Expression too large to display}\) | \(4108\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1680 vs.
\(2 (178) = 356\).
time = 0.39, size = 1680, normalized size = 8.62 \begin {gather*} \frac {3 \, \sqrt {c x^{2} + a} c^{5} d^{5}}{8 \, {\left (c^{4} d^{8} e^{3} + 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} + 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c^{5} d^{4} x}{8 \, {\left (c^{4} d^{8} e^{2} + 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} + 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{4} d^{4}}{8 \, {\left (c^{4} d^{8} x e^{2} + c^{4} d^{9} e + 4 \, a c^{3} d^{6} x e^{4} + 4 \, a c^{3} d^{7} e^{3} + 6 \, a^{2} c^{2} d^{4} x e^{6} + 6 \, a^{2} c^{2} d^{5} e^{5} + 4 \, a^{3} c d^{2} x e^{8} + 4 \, a^{3} c d^{3} e^{7} + a^{4} x e^{10} + a^{4} d e^{9}\right )}} + \frac {3 \, c^{5} d^{5} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-11\right )}}{8 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{3} d^{3}}{8 \, {\left (c^{4} d^{8} x^{2} e + c^{4} d^{10} e^{\left (-1\right )} + 2 \, c^{4} d^{9} x + 4 \, a c^{3} d^{6} x^{2} e^{3} + 8 \, a c^{3} d^{7} x e^{2} + 4 \, a c^{3} d^{8} e + 6 \, a^{2} c^{2} d^{4} x^{2} e^{5} + 12 \, a^{2} c^{2} d^{5} x e^{4} + 6 \, a^{2} c^{2} d^{6} e^{3} + 4 \, a^{3} c d^{2} x^{2} e^{7} + 8 \, a^{3} c d^{3} x e^{6} + 4 \, a^{3} c d^{4} e^{5} + a^{4} x^{2} e^{9} + 2 \, a^{4} d x e^{8} + a^{4} d^{2} e^{7}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{4} d^{3}}{8 \, {\left (c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )}} - \frac {3 \, \sqrt {c x^{2} + a} c^{4} d^{3}}{4 \, {\left (c^{3} d^{6} e^{3} + 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} + a^{3} e^{9}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c^{4} d^{2} x}{8 \, {\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )}} - \frac {3 \, c^{4} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-9\right )}}{4 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {5}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d^{2}}{4 \, {\left (c^{3} d^{6} x^{3} e^{2} + 3 \, c^{3} d^{7} x^{2} e + c^{3} d^{9} e^{\left (-1\right )} + 3 \, c^{3} d^{8} x + 3 \, a c^{2} d^{4} x^{3} e^{4} + 9 \, a c^{2} d^{5} x^{2} e^{3} + 9 \, a c^{2} d^{6} x e^{2} + 3 \, a c^{2} d^{7} e + 3 \, a^{2} c d^{2} x^{3} e^{6} + 9 \, a^{2} c d^{3} x^{2} e^{5} + 9 \, a^{2} c d^{4} x e^{4} + 3 \, a^{2} c d^{5} e^{3} + a^{3} x^{3} e^{8} + 3 \, a^{3} d x^{2} e^{7} + 3 \, a^{3} d^{2} x e^{6} + a^{3} d^{3} e^{5}\right )}} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d^{2}}{8 \, {\left (c^{3} d^{6} x e^{2} + c^{3} d^{7} e + 3 \, a c^{2} d^{4} x e^{4} + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{2} x e^{6} + 3 \, a^{2} c d^{3} e^{5} + a^{3} x e^{8} + a^{3} d e^{7}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c^{2} d}{8 \, {\left (c^{3} d^{6} x^{2} e + c^{3} d^{8} e^{\left (-1\right )} + 2 \, c^{3} d^{7} x + 3 \, a c^{2} d^{4} x^{2} e^{3} + 6 \, a c^{2} d^{5} x e^{2} + 3 \, a c^{2} d^{6} e + 3 \, a^{2} c d^{2} x^{2} e^{5} + 6 \, a^{2} c d^{3} x e^{4} + 3 \, a^{2} c d^{4} e^{3} + a^{3} x^{2} e^{7} + 2 \, a^{3} d x e^{6} + a^{3} d^{2} e^{5}\right )}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3} d}{8 \, {\left (c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )}} + \frac {3 \, c^{3} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{8 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} c d}{4 \, {\left (c^{2} d^{4} x^{4} e^{3} + 4 \, c^{2} d^{5} x^{3} e^{2} + 6 \, c^{2} d^{6} x^{2} e + c^{2} d^{8} e^{\left (-1\right )} + 4 \, c^{2} d^{7} x + 2 \, a c d^{2} x^{4} e^{5} + 8 \, a c d^{3} x^{3} e^{4} + 12 \, a c d^{4} x^{2} e^{3} + 8 \, a c d^{5} x e^{2} + 2 \, a c d^{6} e + a^{2} x^{4} e^{7} + 4 \, a^{2} d x^{3} e^{6} + 6 \, a^{2} d^{2} x^{2} e^{5} + 4 \, a^{2} d^{3} x e^{4} + a^{2} d^{4} e^{3}\right )}} + \frac {3 \, \sqrt {c x^{2} + a} c^{3} d}{8 \, {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{5 \, {\left (c d^{2} x^{5} e^{4} + 5 \, c d^{3} x^{4} e^{3} + 10 \, c d^{4} x^{3} e^{2} + 10 \, c d^{5} x^{2} e + c d^{7} e^{\left (-1\right )} + 5 \, c d^{6} x + a x^{5} e^{6} + 5 \, a d x^{4} e^{5} + 10 \, a d^{2} x^{3} e^{4} + 10 \, a d^{3} x^{2} e^{3} + 5 \, a d^{4} x e^{2} + a d^{5} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 813 vs.
\(2 (178) = 356\).
time = 7.93, size = 1653, normalized size = 8.48 \begin {gather*} \left [\frac {15 \, {\left (a^{2} c^{3} d x^{5} e^{5} + 5 \, a^{2} c^{3} d^{2} x^{4} e^{4} + 10 \, a^{2} c^{3} d^{3} x^{3} e^{3} + 10 \, a^{2} c^{3} d^{4} x^{2} e^{2} + 5 \, a^{2} c^{3} d^{5} x e + a^{2} c^{3} d^{6}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (10 \, c^{5} d^{7} x^{3} + 25 \, a c^{4} d^{7} x - 8 \, {\left (a^{3} c^{2} x^{4} + 2 \, a^{4} c x^{2} + a^{5}\right )} e^{7} - 5 \, {\left (5 \, a^{3} c^{2} d x^{3} + 2 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} - 93 \, a^{3} c^{2} d^{2} x^{2} - 34 \, a^{4} c d^{2}\right )} e^{5} + 5 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 11 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} - 48 \, a^{2} c^{3} d^{4} x^{2} - 59 \, a^{3} c^{2} d^{4}\right )} e^{3} + 5 \, {\left (11 \, a c^{4} d^{5} x^{3} - 4 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 29 \, a c^{4} d^{6} x^{2} - 33 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{80 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} + 2 \, {\left (2 \, a^{3} c d^{2} x^{5} + 5 \, a^{4} d^{2} x^{3}\right )} e^{11} + 10 \, {\left (2 \, a^{3} c d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} + 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} + 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} + 4 \, {\left (a c^{3} d^{6} x^{5} + 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} + 4 \, {\left (5 \, a c^{3} d^{7} x^{4} + 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} + 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} + 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} + 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} + 2 \, a c^{3} d^{11}\right )} e^{2}\right )}}, \frac {15 \, {\left (a^{2} c^{3} d x^{5} e^{5} + 5 \, a^{2} c^{3} d^{2} x^{4} e^{4} + 10 \, a^{2} c^{3} d^{3} x^{3} e^{3} + 10 \, a^{2} c^{3} d^{4} x^{2} e^{2} + 5 \, a^{2} c^{3} d^{5} x e + a^{2} c^{3} d^{6}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) + {\left (10 \, c^{5} d^{7} x^{3} + 25 \, a c^{4} d^{7} x - 8 \, {\left (a^{3} c^{2} x^{4} + 2 \, a^{4} c x^{2} + a^{5}\right )} e^{7} - 5 \, {\left (5 \, a^{3} c^{2} d x^{3} + 2 \, a^{4} c d x\right )} e^{6} + {\left (a^{2} c^{3} d^{2} x^{4} - 93 \, a^{3} c^{2} d^{2} x^{2} - 34 \, a^{4} c d^{2}\right )} e^{5} + 5 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 11 \, a^{3} c^{2} d^{3} x\right )} e^{4} + {\left (11 \, a c^{4} d^{4} x^{4} - 48 \, a^{2} c^{3} d^{4} x^{2} - 59 \, a^{3} c^{2} d^{4}\right )} e^{3} + 5 \, {\left (11 \, a c^{4} d^{5} x^{3} - 4 \, a^{2} c^{3} d^{5} x\right )} e^{2} + {\left (2 \, c^{5} d^{6} x^{4} + 29 \, a c^{4} d^{6} x^{2} - 33 \, a^{2} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + a}}{40 \, {\left (5 \, c^{4} d^{12} x e + c^{4} d^{13} + a^{4} x^{5} e^{13} + 5 \, a^{4} d x^{4} e^{12} + 2 \, {\left (2 \, a^{3} c d^{2} x^{5} + 5 \, a^{4} d^{2} x^{3}\right )} e^{11} + 10 \, {\left (2 \, a^{3} c d^{3} x^{4} + a^{4} d^{3} x^{2}\right )} e^{10} + {\left (6 \, a^{2} c^{2} d^{4} x^{5} + 40 \, a^{3} c d^{4} x^{3} + 5 \, a^{4} d^{4} x\right )} e^{9} + {\left (30 \, a^{2} c^{2} d^{5} x^{4} + 40 \, a^{3} c d^{5} x^{2} + a^{4} d^{5}\right )} e^{8} + 4 \, {\left (a c^{3} d^{6} x^{5} + 15 \, a^{2} c^{2} d^{6} x^{3} + 5 \, a^{3} c d^{6} x\right )} e^{7} + 4 \, {\left (5 \, a c^{3} d^{7} x^{4} + 15 \, a^{2} c^{2} d^{7} x^{2} + a^{3} c d^{7}\right )} e^{6} + {\left (c^{4} d^{8} x^{5} + 40 \, a c^{3} d^{8} x^{3} + 30 \, a^{2} c^{2} d^{8} x\right )} e^{5} + {\left (5 \, c^{4} d^{9} x^{4} + 40 \, a c^{3} d^{9} x^{2} + 6 \, a^{2} c^{2} d^{9}\right )} e^{4} + 10 \, {\left (c^{4} d^{10} x^{3} + 2 \, a c^{3} d^{10} x\right )} e^{3} + 2 \, {\left (5 \, c^{4} d^{11} x^{2} + 2 \, a c^{3} d^{11}\right )} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1246 vs.
\(2 (178) = 356\).
time = 0.84, size = 1246, normalized size = 6.39 \begin {gather*} -\frac {3 \, a^{2} c^{3} d \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{4 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} c^{\frac {13}{2}} d^{8} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{7} d^{9} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} c^{6} d^{7} e^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} c^{\frac {11}{2}} d^{6} e^{3} - 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {13}{2}} d^{8} e - 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{6} d^{7} e^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a c^{\frac {11}{2}} d^{6} e^{3} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a c^{5} d^{5} e^{4} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{6} d^{7} e^{2} + 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a c^{\frac {9}{2}} d^{4} e^{5} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {11}{2}} d^{6} e^{3} - 788 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{2} c^{5} d^{5} e^{4} - 530 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{2} c^{\frac {9}{2}} d^{4} e^{5} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {11}{2}} d^{6} e^{3} - 230 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a^{2} c^{4} d^{3} e^{6} + 400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{3} c^{5} d^{5} e^{4} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a^{2} c^{\frac {7}{2}} d^{2} e^{7} + 1170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{3} c^{\frac {9}{2}} d^{4} e^{5} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} a^{2} c^{3} d e^{8} + 910 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{3} c^{4} d^{3} e^{6} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{4} c^{5} d^{5} e^{4} + 570 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{3} c^{\frac {7}{2}} d^{2} e^{7} - 230 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{4} c^{\frac {9}{2}} d^{4} e^{5} + 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a^{3} c^{3} d e^{8} - 770 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{4} c^{4} d^{3} e^{6} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} a^{3} c^{\frac {5}{2}} e^{9} - 480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{4} c^{\frac {7}{2}} d^{2} e^{7} - 2 \, a^{5} c^{\frac {9}{2}} d^{4} e^{5} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{4} c^{3} d e^{8} + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{5} c^{4} d^{3} e^{6} + 350 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{5} c^{\frac {7}{2}} d^{2} e^{7} + 170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{5} c^{3} d e^{8} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{5} c^{\frac {5}{2}} e^{9} - 9 \, a^{6} c^{\frac {7}{2}} d^{2} e^{7} - 65 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{6} c^{3} d e^{8} + 8 \, a^{7} c^{\frac {5}{2}} e^{9}}{20 \, {\left (c^{3} d^{6} e^{4} + 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} + a^{3} e^{10}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________