Optimal. Leaf size=139 \[ -\frac {2 (B d-A e)}{e (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)}-\frac {4 \sqrt {d+e x} (a B e-4 A b e+3 b B d)}{3 \sqrt {a+b x} (b d-a e)^3}+\frac {2 \sqrt {d+e x} (a B e-4 A b e+3 b B d)}{3 e (a+b x)^{3/2} (b d-a e)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \begin {gather*} -\frac {2 (B d-A e)}{e (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)}-\frac {4 \sqrt {d+e x} (a B e-4 A b e+3 b B d)}{3 \sqrt {a+b x} (b d-a e)^3}+\frac {2 \sqrt {d+e x} (a B e-4 A b e+3 b B d)}{3 e (a+b x)^{3/2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} (d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt {d+e x}}-\frac {(3 b B d-4 A b e+a B e) \int \frac {1}{(a+b x)^{5/2} \sqrt {d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt {d+e x}}+\frac {2 (3 b B d-4 A b e+a B e) \sqrt {d+e x}}{3 e (b d-a e)^2 (a+b x)^{3/2}}+\frac {(2 (3 b B d-4 A b e+a B e)) \int \frac {1}{(a+b x)^{3/2} \sqrt {d+e x}} \, dx}{3 (b d-a e)^2}\\ &=-\frac {2 (B d-A e)}{e (b d-a e) (a+b x)^{3/2} \sqrt {d+e x}}+\frac {2 (3 b B d-4 A b e+a B e) \sqrt {d+e x}}{3 e (b d-a e)^2 (a+b x)^{3/2}}-\frac {4 (3 b B d-4 A b e+a B e) \sqrt {d+e x}}{3 (b d-a e)^3 \sqrt {a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 125, normalized size = 0.90 \begin {gather*} -\frac {2 \left (A \left (-3 a^2 e^2-6 a b e (d+2 e x)+b^2 \left (d^2-4 d e x-8 e^2 x^2\right )\right )+B \left (3 a^2 e (2 d+e x)+2 a b \left (d^2+5 d e x+e^2 x^2\right )+3 b^2 d x (d+2 e x)\right )\right )}{3 (a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.16, size = 133, normalized size = 0.96 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (\frac {3 A e^2 (a+b x)^2}{(d+e x)^2}+\frac {6 A b e (a+b x)}{d+e x}-\frac {3 b B d (a+b x)}{d+e x}-\frac {3 a B e (a+b x)}{d+e x}-\frac {3 B d e (a+b x)^2}{(d+e x)^2}+a b B-A b^2\right )}{3 (a+b x)^{3/2} (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 6.14, size = 337, normalized size = 2.42 \begin {gather*} \frac {2 \, {\left (3 \, A a^{2} e^{2} - {\left (2 \, B a b + A b^{2}\right )} d^{2} - 6 \, {\left (B a^{2} - A a b\right )} d e - 2 \, {\left (3 \, B b^{2} d e + {\left (B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} - {\left (3 \, B b^{2} d^{2} + 2 \, {\left (5 \, B a b - 2 \, A b^{2}\right )} d e + 3 \, {\left (B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.51, size = 565, normalized size = 4.06 \begin {gather*} -\frac {2 \, {\left (B b^{2} d e - A b^{2} e^{2}\right )} \sqrt {b x + a}}{{\left (b^{3} d^{3} {\left | b \right |} - 3 \, a b^{2} d^{2} {\left | b \right |} e + 3 \, a^{2} b d {\left | b \right |} e^{2} - a^{3} {\left | b \right |} e^{3}\right )} \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} - \frac {4 \, {\left (3 \, B b^{\frac {13}{2}} d^{3} e^{\frac {1}{2}} - 4 \, B a b^{\frac {11}{2}} d^{2} e^{\frac {3}{2}} - 5 \, A b^{\frac {13}{2}} d^{2} e^{\frac {3}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac {9}{2}} d^{2} e^{\frac {1}{2}} - B a^{2} b^{\frac {9}{2}} d e^{\frac {5}{2}} + 10 \, A a b^{\frac {11}{2}} d e^{\frac {5}{2}} + 12 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac {9}{2}} d e^{\frac {3}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B b^{\frac {5}{2}} d e^{\frac {1}{2}} + 2 \, B a^{3} b^{\frac {7}{2}} e^{\frac {7}{2}} - 5 \, A a^{2} b^{\frac {9}{2}} e^{\frac {7}{2}} + 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a^{2} b^{\frac {5}{2}} e^{\frac {5}{2}} - 12 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A a b^{\frac {7}{2}} e^{\frac {5}{2}} - 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} A b^{\frac {5}{2}} e^{\frac {3}{2}}\right )}}{3 \, {\left (b^{2} d^{2} {\left | b \right |} - 2 \, a b d {\left | b \right |} e + a^{2} {\left | b \right |} e^{2}\right )} {\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 177, normalized size = 1.27 \begin {gather*} -\frac {2 \left (8 A \,b^{2} e^{2} x^{2}-2 B a b \,e^{2} x^{2}-6 B \,b^{2} d e \,x^{2}+12 A a b \,e^{2} x +4 A \,b^{2} d e x -3 B \,a^{2} e^{2} x -10 B a b d e x -3 B \,b^{2} d^{2} x +3 A \,a^{2} e^{2}+6 A a b d e -A \,b^{2} d^{2}-6 B \,a^{2} d e -2 B a b \,d^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \sqrt {e x +d}\, \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\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 = 2.26, size = 193, normalized size = 1.39 \begin {gather*} \frac {\sqrt {d+e\,x}\,\left (\frac {4\,x^2\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {12\,B\,a^2\,d\,e-6\,A\,a^2\,e^2+4\,B\,a\,b\,d^2-12\,A\,a\,b\,d\,e+2\,A\,b^2\,d^2}{3\,b\,e\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,x\,\left (3\,a\,e+b\,d\right )\,\left (B\,a\,e-4\,A\,b\,e+3\,B\,b\,d\right )}{3\,b\,e\,{\left (a\,e-b\,d\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a\,d\,\sqrt {a+b\,x}}{b\,e}+\frac {x\,\left (a\,e+b\,d\right )\,\sqrt {a+b\,x}}{b\,e}} \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]
________________________________________________________________________________________