Optimal. Leaf size=209 \[ -\frac {e^2 (-a B e-5 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{7/2}}+\frac {e \sqrt {d+e x} (-a B e-5 A b e+6 b B d)}{8 b (a+b x) (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-5 A b e+6 b B d)}{12 b (a+b x)^2 (b d-a e)^2}-\frac {\sqrt {d+e x} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \begin {gather*} -\frac {e^2 (-a B e-5 A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{7/2}}+\frac {e \sqrt {d+e x} (-a B e-5 A b e+6 b B d)}{8 b (a+b x) (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-5 A b e+6 b B d)}{12 b (a+b x)^2 (b d-a e)^2}-\frac {\sqrt {d+e x} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
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 )^2} \, dx &=\int \frac {A+B x}{(a+b x)^4 \sqrt {d+e x}} \, dx\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^3}+\frac {(6 b B d-5 A b e-a B e) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{6 b (b d-a e)}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^3}-\frac {(6 b B d-5 A b e-a B e) \sqrt {d+e x}}{12 b (b d-a e)^2 (a+b x)^2}-\frac {(e (6 b B d-5 A b e-a B e)) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^3}-\frac {(6 b B d-5 A b e-a B e) \sqrt {d+e x}}{12 b (b d-a e)^2 (a+b x)^2}+\frac {e (6 b B d-5 A b e-a B e) \sqrt {d+e x}}{8 b (b d-a e)^3 (a+b x)}+\frac {\left (e^2 (6 b B d-5 A b e-a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b (b d-a e)^3}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^3}-\frac {(6 b B d-5 A b e-a B e) \sqrt {d+e x}}{12 b (b d-a e)^2 (a+b x)^2}+\frac {e (6 b B d-5 A b e-a B e) \sqrt {d+e x}}{8 b (b d-a e)^3 (a+b x)}+\frac {(e (6 b B d-5 A b e-a B e)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b (b d-a e)^3}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^3}-\frac {(6 b B d-5 A b e-a B e) \sqrt {d+e x}}{12 b (b d-a e)^2 (a+b x)^2}+\frac {e (6 b B d-5 A b e-a B e) \sqrt {d+e x}}{8 b (b d-a e)^3 (a+b x)}-\frac {e^2 (6 b B d-5 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{3/2} (b d-a e)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 97, normalized size = 0.46 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {e^2 (a B e+5 A b e-6 b B d) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}+\frac {a B-A b}{(a+b x)^3}\right )}{3 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.89, size = 325, normalized size = 1.56 \begin {gather*} \frac {e^2 \sqrt {d+e x} \left (-3 a^3 B e^3+33 a^2 A b e^3+8 a^2 b B e^2 (d+e x)-24 a^2 b B d e^2+40 a A b^2 e^2 (d+e x)-66 a A b^2 d e^2+57 a b^2 B d^2 e+3 a b^2 B e (d+e x)^2-56 a b^2 B d e (d+e x)+33 A b^3 d^2 e+15 A b^3 e (d+e x)^2-40 A b^3 d e (d+e x)-30 b^3 B d^3+48 b^3 B d^2 (d+e x)-18 b^3 B d (d+e x)^2\right )}{24 b (b d-a e)^3 (-a e-b (d+e x)+b d)^3}+\frac {\left (a B e^3+5 A b e^3-6 b B d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 b^{3/2} (b d-a e)^3 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 1337, normalized size = 6.40
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 429, normalized size = 2.05 \begin {gather*} \frac {{\left (6 \, B b d e^{2} - B a e^{3} - 5 \, A b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {18 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 48 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{2} + 30 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} - 15 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 56 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 57 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} - 33 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} + 24 \, \sqrt {x e + d} B a^{2} b d e^{4} + 66 \, \sqrt {x e + d} A a b^{2} d e^{4} + 3 \, \sqrt {x e + d} B a^{3} e^{5} - 33 \, \sqrt {x e + d} A a^{2} b e^{5}}{24 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 679, normalized size = 3.25 \begin {gather*} \frac {5 \left (e x +d \right )^{\frac {5}{2}} A \,b^{2} e^{3}}{8 \left (b e x +a e \right )^{3} \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 {\left (e x +d \right )^{\frac {5}{2}} B a b \,e^{3}}{8 \left (b e x +a e \right )^{3} \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 {3 \left (e x +d \right )^{\frac {5}{2}} B \,b^{2} d \,e^{2}}{4 \left (b e x +a e \right )^{3} \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 {5 \left (e x +d \right )^{\frac {3}{2}} A b \,e^{3}}{3 \left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {5 A \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}}+\frac {B a \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}\, b}+\frac {\left (e x +d \right )^{\frac {3}{2}} B a \,e^{3}}{3 \left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B b d \,e^{2}}{\left (b e x +a e \right )^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {3 B d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (a e -b d \right ) b}}+\frac {11 \sqrt {e x +d}\, A \,e^{3}}{8 \left (b e x +a e \right )^{3} \left (a e -b d \right )}-\frac {\sqrt {e x +d}\, B a \,e^{3}}{8 \left (b e x +a e \right )^{3} \left (a e -b d \right ) b}-\frac {5 \sqrt {e x +d}\, B d \,e^{2}}{4 \left (b e x +a e \right )^{3} \left (a e -b d \right )} \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 = 2.13, size = 331, normalized size = 1.58 \begin {gather*} \frac {\frac {{\left (d+e\,x\right )}^{3/2}\,\left (5\,A\,b\,e^3+B\,a\,e^3-6\,B\,b\,d\,e^2\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {b\,{\left (d+e\,x\right )}^{5/2}\,\left (5\,A\,b\,e^3+B\,a\,e^3-6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (B\,a\,e^3-11\,A\,b\,e^3+10\,B\,b\,d\,e^2\right )}{8\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )+b^3\,{\left (d+e\,x\right )}^3-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^2+a^3\,e^3-b^3\,d^3+3\,a\,b^2\,d^2\,e-3\,a^2\,b\,d\,e^2}+\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (5\,A\,b\,e+B\,a\,e-6\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (5\,A\,b\,e^3+B\,a\,e^3-6\,B\,b\,d\,e^2\right )}\right )\,\left (5\,A\,b\,e+B\,a\,e-6\,B\,b\,d\right )}{8\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{7/2}} \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]
________________________________________________________________________________________