Optimal. Leaf size=286 \[ \frac {2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^3 \sqrt {x} (a+b x) (A b-a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \begin {gather*} \frac {2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^3 \sqrt {x} (a+b x) (A b-a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {x^{7/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{7/2}}{a b+b^2 x} \, dx}{9 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{a b+b^2 x} \, dx}{9 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 a^2 \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{a b+b^2 x} \, dx}{9 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a^3 \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{9 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x} (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 a^4 \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{9 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x} (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 a^4 \left (\frac {9 A b^2}{2}-\frac {9 a b B}{2}\right ) \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 )}{9 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a^3 (A b-a B) \sqrt {x} (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 (A b-a B) x^{3/2} (a+b x)}{3 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{5/2} (a+b x)}{5 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{7/2} (a+b x)}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{9/2} (a+b x)}{9 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^{7/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 139, normalized size = 0.49 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )-315 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{315 b^{11/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 17.42, size = 153, normalized size = 0.53 \begin {gather*} \frac {(a+b x) \left (\frac {2 \sqrt {x} \left (315 a^4 B-315 a^3 A b-105 a^3 b B x+105 a^2 A b^2 x+63 a^2 b^2 B x^2-63 a A b^3 x^2-45 a b^3 B x^3+45 A b^4 x^3+35 b^4 B x^4\right )}{315 b^5}-\frac {2 \left (a^{9/2} B-a^{7/2} A b\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 276, normalized size = 0.97 \begin {gather*} \left [-\frac {315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{315 \, b^{5}}, -\frac {2 \, {\left (315 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt {x}\right )}}{315 \, b^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 205, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (B a^{5} \mathrm {sgn}\left (b x + a\right ) - A a^{4} b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (35 \, B b^{8} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) - 45 \, B a b^{7} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 45 \, A b^{8} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a^{2} b^{6} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) - 63 \, A a b^{7} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{6} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 315 \, B a^{4} b^{4} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - 315 \, A a^{3} b^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right )\right )}}{315 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 197, normalized size = 0.69 \begin {gather*} \frac {2 \left (b x +a \right ) \left (35 \sqrt {a b}\, B \,b^{4} x^{\frac {9}{2}}+45 \sqrt {a b}\, A \,b^{4} x^{\frac {7}{2}}-45 \sqrt {a b}\, B a \,b^{3} x^{\frac {7}{2}}-63 \sqrt {a b}\, A a \,b^{3} x^{\frac {5}{2}}+63 \sqrt {a b}\, B \,a^{2} b^{2} x^{\frac {5}{2}}+315 A \,a^{4} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-315 B \,a^{5} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 \sqrt {a b}\, A \,a^{2} b^{2} x^{\frac {3}{2}}-105 \sqrt {a b}\, B \,a^{3} b \,x^{\frac {3}{2}}-315 \sqrt {a b}\, A \,a^{3} b \sqrt {x}+315 \sqrt {a b}\, B \,a^{4} \sqrt {x}\right )}{315 \sqrt {\left (b x +a \right )^{2}}\, \sqrt {a b}\, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.52, size = 257, normalized size = 0.90 \begin {gather*} \frac {10 \, {\left (7 \, B b^{4} x^{2} + 9 \, B a b^{3} x\right )} x^{\frac {7}{2}} - 2 \, {\left (5 \, {\left (11 \, B a b^{3} - 9 \, A b^{4}\right )} x^{2} + 9 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x\right )} x^{\frac {5}{2}} + 6 \, {\left (3 \, {\left (11 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{2} + 7 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 21 \, {\left (3 \, {\left (11 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {x}}{315 \, {\left (b^{5} x + a b^{4}\right )}} - \frac {2 \, {\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} - \frac {{\left (11 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{4} - A a^{3} b\right )} \sqrt {x}}{3 \, b^{5}} \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 {x^{7/2}\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^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]
________________________________________________________________________________________