Optimal. Leaf size=313 \[ \frac {e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac {e^3 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac {e^2 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac {e \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 51, 63, 208} \begin {gather*} -\frac {e^3 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac {e^2 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}+\frac {e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac {e \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 47
Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-7 A b e-3 a B e) \int \frac {\sqrt {d+e x}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-7 A b e-3 a B e)) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^2 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b^2 (b d-a e)^2}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^2 (b d-a e)^3}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^2 (b d-a e)^4}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^2 (b d-a e)^4}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 99, normalized size = 0.32 \begin {gather*} \frac {(d+e x)^{3/2} \left (\frac {e^4 (3 a B e+7 A b e-10 b B d) \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {3 a B-3 A b}{(a+b x)^5}\right )}{15 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 2.72, size = 676, normalized size = 2.16 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (45 a^5 B e^5+105 a^4 A b e^5+210 a^4 b B e^4 (d+e x)-330 a^4 b B d e^4-790 a^3 A b^2 e^4 (d+e x)-420 a^3 A b^2 d e^4+870 a^3 b^2 B d^2 e^3-384 a^3 b^2 B e^3 (d+e x)^2-50 a^3 b^2 B d e^3 (d+e x)+630 a^2 A b^3 d^2 e^3-896 a^2 A b^3 e^3 (d+e x)^2+2370 a^2 A b^3 d e^3 (d+e x)-1080 a^2 b^3 B d^3 e^2-1110 a^2 b^3 B d^2 e^2 (d+e x)-210 a^2 b^3 B e^2 (d+e x)^3+2048 a^2 b^3 B d e^2 (d+e x)^2-420 a A b^4 d^3 e^2-2370 a A b^4 d^2 e^2 (d+e x)-490 a A b^4 e^2 (d+e x)^3+1792 a A b^4 d e^2 (d+e x)^2+645 a b^4 B d^4 e+1530 a b^4 B d^3 e (d+e x)-2944 a b^4 B d^2 e (d+e x)^2-45 a b^4 B e (d+e x)^4+910 a b^4 B d e (d+e x)^3+105 A b^5 d^4 e+790 A b^5 d^3 e (d+e x)-896 A b^5 d^2 e (d+e x)^2-105 A b^5 e (d+e x)^4+490 A b^5 d e (d+e x)^3-150 b^5 B d^5-580 b^5 B d^4 (d+e x)+1280 b^5 B d^3 (d+e x)^2-700 b^5 B d^2 (d+e x)^3+150 b^5 B d (d+e x)^4\right )}{1920 b^2 (b d-a e)^4 (-a e-b (d+e x)+b d)^5}+\frac {\left (-3 a B e^5-7 A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{5/2} (b d-a e)^4 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 2580, normalized size = 8.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.27, size = 857, normalized size = 2.74 \begin {gather*} -\frac {{\left (10 \, B b d e^{4} - 3 \, B a e^{5} - 7 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 700 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 1280 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 580 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 45 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 910 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 2944 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 790 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 645 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 105 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 210 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 2048 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 1110 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1080 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 420 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 384 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 50 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 870 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 630 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 790 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 330 \, \sqrt {x e + d} B a^{4} b d e^{8} - 420 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 45 \, \sqrt {x e + d} B a^{5} e^{9} + 105 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.09, size = 1037, normalized size = 3.31 \begin {gather*} \frac {7 \left (e x +d \right )^{\frac {9}{2}} A \,b^{3} e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {3 \left (e x +d \right )^{\frac {9}{2}} B a \,b^{2} e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {5 \left (e x +d \right )^{\frac {9}{2}} B \,b^{3} d \,e^{4}}{64 \left (b e x +a e \right )^{5} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {49 \left (e x +d \right )^{\frac {7}{2}} A \,b^{2} e^{5}}{192 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} B a b \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {35 \left (e x +d \right )^{\frac {7}{2}} B \,b^{2} d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {7 \left (e x +d \right )^{\frac {5}{2}} A b \,e^{5}}{15 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (e x +d \right )^{\frac {5}{2}} B a \,e^{5}}{5 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} B b d \,e^{4}}{3 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 A \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, b}+\frac {79 \left (e x +d \right )^{\frac {3}{2}} A \,e^{5}}{192 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} B a \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a e -b d \right ) b}+\frac {3 B a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {5 B d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, b}-\frac {29 \left (e x +d \right )^{\frac {3}{2}} B d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {7 \sqrt {e x +d}\, A \,e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {3 \sqrt {e x +d}\, B a \,e^{5}}{128 \left (b e x +a e \right )^{5} b^{2}}+\frac {5 \sqrt {e x +d}\, B d \,e^{4}}{64 \left (b e x +a e \right )^{5} b} \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 [B] time = 2.24, size = 564, normalized size = 1.80 \begin {gather*} \frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (7\,A\,b^2\,e^5-10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^2}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {b^2\,{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (21\,B\,a\,e^5-79\,A\,b\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________