Optimal. Leaf size=231 \[ \frac {5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}+\frac {5 e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}-\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {668, 640, 621, 206} \begin {gather*} -\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {5 e^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3}+\frac {5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}-\frac {2 (d+e x)^4}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 640
Rule 668
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(5 e) \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac {2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 e^2\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c^2 d^2}\\ &=-\frac {2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac {\left (5 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^3 d^3}\\ &=-\frac {2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac {\left (5 e^2 \left (c d^2-a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^3 d^3}\\ &=-\frac {2 (d+e x)^4}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 e (d+e x)^2}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3}+\frac {5 e^{3/2} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 c^{7/2} d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 112, normalized size = 0.48 \begin {gather*} -\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{3 c^3 d^3 (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.40, size = 641, normalized size = 2.77 \begin {gather*} \left [\frac {15 \, {\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} + {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) + 4 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \, {\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{12 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}, -\frac {15 \, {\left (a^{2} c d^{2} e^{3} - a^{3} e^{5} + {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} - 10 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} - 2 \, {\left (7 \, c^{2} d^{3} e - 10 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{6 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.77, size = 827, normalized size = 3.58 \begin {gather*} \frac {{\left ({\left ({\left (\frac {3 \, {\left (c^{6} d^{10} e^{6} - 4 \, a c^{5} d^{8} e^{8} + 6 \, a^{2} c^{4} d^{6} e^{10} - 4 \, a^{3} c^{3} d^{4} e^{12} + a^{4} c^{2} d^{2} e^{14}\right )} x}{c^{7} d^{11} e^{2} - 4 \, a c^{6} d^{9} e^{4} + 6 \, a^{2} c^{5} d^{7} e^{6} - 4 \, a^{3} c^{4} d^{5} e^{8} + a^{4} c^{3} d^{3} e^{10}} - \frac {4 \, {\left (2 \, c^{6} d^{11} e^{5} - 13 \, a c^{5} d^{9} e^{7} + 32 \, a^{2} c^{4} d^{7} e^{9} - 38 \, a^{3} c^{3} d^{5} e^{11} + 22 \, a^{4} c^{2} d^{3} e^{13} - 5 \, a^{5} c d e^{15}\right )}}{c^{7} d^{11} e^{2} - 4 \, a c^{6} d^{9} e^{4} + 6 \, a^{2} c^{5} d^{7} e^{6} - 4 \, a^{3} c^{4} d^{5} e^{8} + a^{4} c^{3} d^{3} e^{10}}\right )} x - \frac {3 \, {\left (9 \, c^{6} d^{12} e^{4} - 46 \, a c^{5} d^{10} e^{6} + 89 \, a^{2} c^{4} d^{8} e^{8} - 76 \, a^{3} c^{3} d^{6} e^{10} + 19 \, a^{4} c^{2} d^{4} e^{12} + 10 \, a^{5} c d^{2} e^{14} - 5 \, a^{6} e^{16}\right )}}{c^{7} d^{11} e^{2} - 4 \, a c^{6} d^{9} e^{4} + 6 \, a^{2} c^{5} d^{7} e^{6} - 4 \, a^{3} c^{4} d^{5} e^{8} + a^{4} c^{3} d^{3} e^{10}}\right )} x - \frac {6 \, {\left (3 \, c^{6} d^{13} e^{3} - 12 \, a c^{5} d^{11} e^{5} + 13 \, a^{2} c^{4} d^{9} e^{7} + 8 \, a^{3} c^{3} d^{7} e^{9} - 27 \, a^{4} c^{2} d^{5} e^{11} + 20 \, a^{5} c d^{3} e^{13} - 5 \, a^{6} d e^{15}\right )}}{c^{7} d^{11} e^{2} - 4 \, a c^{6} d^{9} e^{4} + 6 \, a^{2} c^{5} d^{7} e^{6} - 4 \, a^{3} c^{4} d^{5} e^{8} + a^{4} c^{3} d^{3} e^{10}}\right )} x - \frac {2 \, c^{6} d^{14} e^{2} + 2 \, a c^{5} d^{12} e^{4} - 43 \, a^{2} c^{4} d^{10} e^{6} + 112 \, a^{3} c^{3} d^{8} e^{8} - 128 \, a^{4} c^{2} d^{6} e^{10} + 70 \, a^{5} c d^{4} e^{12} - 15 \, a^{6} d^{2} e^{14}}{c^{7} d^{11} e^{2} - 4 \, a c^{6} d^{9} e^{4} + 6 \, a^{2} c^{5} d^{7} e^{6} - 4 \, a^{3} c^{4} d^{5} e^{8} + a^{4} c^{3} d^{3} e^{10}}}{3 \, {\left (c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (c d^{2} e^{2} - a e^{4}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{2 \, c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 3215, normalized size = 13.92 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^5}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \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 (d + e x\right )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________