Optimal. Leaf size=313 \[ \frac {e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac {e^3 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac {e^2 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}-\frac {e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.27, 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} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac {e^2 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}+\frac {e^4 (-7 a B e-3 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^{9/2} (b d-a e)^{5/2}}-\frac {e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{7/2} (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) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-3 A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-3 A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^2 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^3} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^4 (b d-a e)^2}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-3 A b e-7 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^4 (b d-a e)^2}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-3 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.06, size = 99, normalized size = 0.32 \begin {gather*} \frac {(d+e x)^{7/2} \left (\frac {e^4 (7 a B e+3 A b e-10 b B d) \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {7 a B-7 A b}{(a+b x)^5}\right )}{35 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 4.64, size = 676, normalized size = 2.16 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (105 a^5 B e^5+45 a^4 A b e^5+490 a^4 b B e^4 (d+e x)-570 a^4 b B d e^4+210 a^3 A b^2 e^4 (d+e x)-180 a^3 A b^2 d e^4+1230 a^3 b^2 B d^2 e^3+896 a^3 b^2 B e^3 (d+e x)^2-2170 a^3 b^2 B d e^3 (d+e x)+270 a^2 A b^3 d^2 e^3+384 a^2 A b^3 e^3 (d+e x)^2-630 a^2 A b^3 d e^3 (d+e x)-1320 a^2 b^3 B d^3 e^2+3570 a^2 b^3 B d^2 e^2 (d+e x)+790 a^2 b^3 B e^2 (d+e x)^3-3072 a^2 b^3 B d e^2 (d+e x)^2-180 a A b^4 d^3 e^2+630 a A b^4 d^2 e^2 (d+e x)-210 a A b^4 e^2 (d+e x)^3-768 a A b^4 d e^2 (d+e x)^2+705 a b^4 B d^4 e-2590 a b^4 B d^3 e (d+e x)+3456 a b^4 B d^2 e (d+e x)^2-105 a b^4 B e (d+e x)^4-1370 a b^4 B d e (d+e x)^3+45 A b^5 d^4 e-210 A b^5 d^3 e (d+e x)+384 A b^5 d^2 e (d+e x)^2-45 A b^5 e (d+e x)^4+210 A b^5 d e (d+e x)^3-150 b^5 B d^5+700 b^5 B d^4 (d+e x)-1280 b^5 B d^3 (d+e x)^2+580 b^5 B d^2 (d+e x)^3+150 b^5 B d (d+e x)^4\right )}{1920 b^4 (b d-a e)^2 (-a e-b (d+e x)+b d)^5}+\frac {\left (-7 a B e^5-3 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^{9/2} (b d-a e)^2 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 2238, normalized size = 7.15
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.30, size = 805, normalized size = 2.57 \begin {gather*} -\frac {{\left (10 \, B b d e^{4} - 7 \, B a e^{5} - 3 \, 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^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} + 580 \, {\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} + 700 \, {\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} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 45 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} - 1370 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 210 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} + 3456 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} + 384 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} - 2590 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 705 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 45 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} + 790 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 210 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} - 3072 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} - 768 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} + 3570 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1320 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 180 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} + 384 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 2170 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 1230 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 270 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 490 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 570 \, \sqrt {x e + d} B a^{4} b d e^{8} - 180 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 105 \, \sqrt {x e + d} B a^{5} e^{9} + 45 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\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 = 0.08, size = 872, normalized size = 2.79 \begin {gather*} -\frac {3 \sqrt {e x +d}\, A \,a^{2} e^{7}}{128 \left (b e x +a e \right )^{5} b^{3}}+\frac {3 \sqrt {e x +d}\, A a d \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {3 \left (e x +d \right )^{\frac {9}{2}} A b \,e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {3 \sqrt {e x +d}\, A \,d^{2} e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {7 \sqrt {e x +d}\, B \,a^{3} e^{7}}{128 \left (b e x +a e \right )^{5} b^{4}}+\frac {3 \sqrt {e x +d}\, B \,a^{2} d \,e^{6}}{16 \left (b e x +a e \right )^{5} b^{3}}-\frac {27 \sqrt {e x +d}\, B a \,d^{2} e^{5}}{128 \left (b e x +a e \right )^{5} b^{2}}+\frac {7 \left (e x +d \right )^{\frac {9}{2}} B a \,e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {5 \left (e x +d \right )^{\frac {9}{2}} B b d \,e^{4}}{64 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {5 \sqrt {e x +d}\, B \,d^{3} e^{4}}{64 \left (b e x +a e \right )^{5} b}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {7 \left (e x +d \right )^{\frac {3}{2}} A d \,e^{5}}{64 \left (b e x +a e \right )^{5} b}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} A \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {49 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{6}}{192 \left (b e x +a e \right )^{5} b^{3}}-\frac {79 \left (e x +d \right )^{\frac {7}{2}} B a \,e^{5}}{192 \left (b e x +a e \right )^{5} \left (a e -b d \right ) b}+\frac {119 \left (e x +d \right )^{\frac {3}{2}} B a d \,e^{5}}{192 \left (b e x +a e \right )^{5} b^{2}}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2} e^{4}}{96 \left (b e x +a e \right )^{5} b}+\frac {29 \left (e x +d \right )^{\frac {7}{2}} B d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {\left (e x +d \right )^{\frac {5}{2}} A \,e^{5}}{5 \left (b e x +a e \right )^{5} b}-\frac {7 \left (e x +d \right )^{\frac {5}{2}} B a \,e^{5}}{15 \left (b e x +a e \right )^{5} b^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B d \,e^{4}}{3 \left (b e x +a e \right )^{5} b}+\frac {3 A \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {7 B a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{4}}-\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^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{3}} \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 = 0.35, size = 572, normalized size = 1.83 \begin {gather*} \frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{9/2}\,{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^2}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^3}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (21\,A\,b\,e^5-79\,B\,a\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}+\frac {\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^4}}{\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} \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]
________________________________________________________________________________________