Optimal. Leaf size=258 \[ \frac {\sqrt {x} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\sqrt {x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \begin {gather*} \frac {\sqrt {x} (A b-a B)}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\sqrt {x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\sqrt {x} \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 b (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )^3} \, dx}{48 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 A b+a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 (7 A b+a B) \sqrt {x}}{64 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(7 A b+a B) \sqrt {x}}{24 a^2 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) \sqrt {x}}{96 a^3 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b+a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 77, normalized size = 0.30 \begin {gather*} \frac {\sqrt {x} \left (a^4 (A b-a B)+(a+b x)^4 (a B+7 A b) \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};-\frac {b x}{a}\right )\right )}{4 a^5 b (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 20.73, size = 152, normalized size = 0.59 \begin {gather*} \frac {(a+b x) \left (\frac {5 (a B+7 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{9/2} b^{3/2}}-\frac {\sqrt {x} \left (15 a^4 B-279 a^3 A b-73 a^3 b B x-511 a^2 A b^2 x-55 a^2 b^2 B x^2-385 a A b^3 x^2-15 a b^3 B x^3-105 A b^4 x^3\right )}{192 a^4 b (a+b x)^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 523, normalized size = 2.03 \begin {gather*} \left [-\frac {15 \, {\left (B a^{5} + 7 \, A a^{4} b + {\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \, {\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \, {\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \, {\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \, {\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \, {\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}, -\frac {15 \, {\left (B a^{5} + 7 \, A a^{4} b + {\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \, {\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \, {\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \, {\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \, {\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \, {\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 147, normalized size = 0.57 \begin {gather*} \frac {5 \, {\left (B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{4} b \mathrm {sgn}\left (b x + a\right )} + \frac {15 \, B a b^{3} x^{\frac {7}{2}} + 105 \, A b^{4} x^{\frac {7}{2}} + 55 \, B a^{2} b^{2} x^{\frac {5}{2}} + 385 \, A a b^{3} x^{\frac {5}{2}} + 73 \, B a^{3} b x^{\frac {3}{2}} + 511 \, A a^{2} b^{2} x^{\frac {3}{2}} - 15 \, B a^{4} \sqrt {x} + 279 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{4} b \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 357, normalized size = 1.38 \begin {gather*} \frac {\left (105 A \,b^{5} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 B a \,b^{4} x^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+420 A a \,b^{4} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+60 B \,a^{2} b^{3} x^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+630 A \,a^{2} b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+90 B \,a^{3} b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 \sqrt {a b}\, A \,b^{4} x^{\frac {7}{2}}+15 \sqrt {a b}\, B a \,b^{3} x^{\frac {7}{2}}+420 A \,a^{3} b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+60 B \,a^{4} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+385 \sqrt {a b}\, A a \,b^{3} x^{\frac {5}{2}}+55 \sqrt {a b}\, B \,a^{2} b^{2} x^{\frac {5}{2}}+105 A \,a^{4} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 B \,a^{5} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+511 \sqrt {a b}\, A \,a^{2} b^{2} x^{\frac {3}{2}}+73 \sqrt {a b}\, B \,a^{3} b \,x^{\frac {3}{2}}+279 \sqrt {a b}\, A \,a^{3} b \sqrt {x}-15 \sqrt {a b}\, B \,a^{4} \sqrt {x}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.87, size = 390, normalized size = 1.51 \begin {gather*} -\frac {5 \, {\left ({\left (3 \, B a b^{5} + 7 \, A b^{6}\right )} x^{2} - 21 \, {\left (B a^{2} b^{4} + 9 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left ({\left (3 \, B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{2} - 63 \, {\left (B a^{3} b^{3} + 9 \, A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} - 20 \, {\left (2 \, {\left (3 \, B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} + 77 \, {\left (B a^{4} b^{2} + 9 \, A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} - 2 \, {\left (85 \, {\left (3 \, B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x^{2} + 973 \, {\left (B a^{5} b + 9 \, A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} - {\left (253 \, {\left (3 \, B a^{5} b + 7 \, A a^{4} b^{2}\right )} x^{2} + 1315 \, {\left (B a^{6} + 9 \, A a^{5} b\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left (A a^{5} b x^{2} + 3 \, A a^{6} x\right )}}{\sqrt {x}}}{1920 \, {\left (a^{6} b^{5} x^{5} + 5 \, a^{7} b^{4} x^{4} + 10 \, a^{8} b^{3} x^{3} + 10 \, a^{9} b^{2} x^{2} + 5 \, a^{10} b x + a^{11}\right )}} + \frac {5 \, {\left (B a + 7 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{4} b} + \frac {{\left (3 \, B a b + 7 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} + 7 \, A a b\right )} \sqrt {x}}{384 \, a^{6} b} \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 {A+B\,x}{\sqrt {x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \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]
________________________________________________________________________________________