Optimal. Leaf size=192 \[ \frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}-\frac {7 a^3 \sqrt {x} \sqrt {a+b x} (10 A b-9 a B)}{128 b^5}+\frac {7 a^2 x^{3/2} \sqrt {a+b x} (10 A b-9 a B)}{192 b^4}-\frac {7 a x^{5/2} \sqrt {a+b x} (10 A b-9 a B)}{240 b^3}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-9 a B)}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \begin {gather*} \frac {7 a^2 x^{3/2} \sqrt {a+b x} (10 A b-9 a B)}{192 b^4}-\frac {7 a^3 \sqrt {x} \sqrt {a+b x} (10 A b-9 a B)}{128 b^5}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-9 a B)}{40 b^2}-\frac {7 a x^{5/2} \sqrt {a+b x} (10 A b-9 a B)}{240 b^3}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\sqrt {a+b x}} \, dx &=\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (5 A b-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{\sqrt {a+b x}} \, dx}{5 b}\\ &=\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {(7 a (10 A b-9 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{80 b^2}\\ &=-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^2 (10 A b-9 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^3}\\ &=\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {\left (7 a^3 (10 A b-9 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^4}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.39, size = 133, normalized size = 0.69 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {(10 A b-9 a B) \left (105 a^{7/2} \sqrt {b} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+b x \sqrt {\frac {b x}{a}+1} \left (-105 a^3+70 a^2 b x-56 a b^2 x^2+48 b^3 x^3\right )\right )}{\sqrt {\frac {b x}{a}+1}}+384 b^5 B x^5\right )}{1920 b^6 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.31, size = 173, normalized size = 0.90 \begin {gather*} \frac {7 \left (9 a^5 B-10 a^4 A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{128 b^{11/2}}+\frac {\sqrt {a+b x} \left (945 a^4 B \sqrt {x}-1050 a^3 A b \sqrt {x}-630 a^3 b B x^{3/2}+700 a^2 A b^2 x^{3/2}+504 a^2 b^2 B x^{5/2}-560 a A b^3 x^{5/2}-432 a b^3 B x^{7/2}+480 A b^4 x^{7/2}+384 b^4 B x^{9/2}\right )}{1920 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.71, size = 296, normalized size = 1.54 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{6}}, \frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [A] time = 0.02, size = 260, normalized size = 1.35 \begin {gather*} \frac {\sqrt {b x +a}\, \left (768 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}+960 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}-864 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}-1120 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}+1008 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+1050 A \,a^{4} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-945 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+1400 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x -1260 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -2100 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+1890 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} \sqrt {b}\right ) \sqrt {x}}{3840 \sqrt {\left (b x +a \right ) x}\, b^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.94, size = 252, normalized size = 1.31 \begin {gather*} \frac {\sqrt {b x^{2} + a x} B x^{4}}{5 \, b} - \frac {9 \, \sqrt {b x^{2} + a x} B a x^{3}}{40 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x^{3}}{4 \, b} + \frac {21 \, \sqrt {b x^{2} + a x} B a^{2} x^{2}}{80 \, b^{3}} - \frac {7 \, \sqrt {b x^{2} + a x} A a x^{2}}{24 \, b^{2}} - \frac {21 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{4}} + \frac {35 \, \sqrt {b x^{2} + a x} A a^{2} x}{96 \, b^{3}} - \frac {63 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} + \frac {63 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{7/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 103.15, size = 360, normalized size = 1.88 \begin {gather*} - \frac {35 A a^{\frac {7}{2}} \sqrt {x}}{64 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {35 A a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 A a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {7}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {35 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {9}{2}}} + \frac {A x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {63 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{5} \sqrt {1 + \frac {b x}{a}}} + \frac {21 B a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {21 B a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {3 B a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {9}{2}}}{40 b \sqrt {1 + \frac {b x}{a}}} - \frac {63 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {11}{2}}} + \frac {B x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________