Optimal. Leaf size=192 \[ -\frac {a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}}+\frac {a^3 \sqrt {x} \sqrt {a+b x} (10 A b-7 a B)}{128 b^4}-\frac {a^2 x^{3/2} \sqrt {a+b x} (10 A b-7 a B)}{192 b^3}+\frac {a x^{5/2} \sqrt {a+b x} (10 A b-7 a B)}{240 b^2}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-7 a B)}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b} \]
________________________________________________________________________________________
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 {a^2 x^{3/2} \sqrt {a+b x} (10 A b-7 a B)}{192 b^3}+\frac {a^3 \sqrt {x} \sqrt {a+b x} (10 A b-7 a B)}{128 b^4}-\frac {a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}}+\frac {a x^{5/2} \sqrt {a+b x} (10 A b-7 a B)}{240 b^2}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-7 a B)}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{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 x^{5/2} \sqrt {a+b x} (A+B x) \, dx &=\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac {\left (5 A b-\frac {7 a B}{2}\right ) \int x^{5/2} \sqrt {a+b x} \, dx}{5 b}\\ &=\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac {(a (10 A b-7 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{80 b}\\ &=\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^2 (10 A b-7 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^2}\\ &=-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}+\frac {\left (a^3 (10 A b-7 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^3}\\ &=\frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 A b-7 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^4}\\ &=\frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 A b-7 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^4}\\ &=\frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {\left (a^4 (10 A b-7 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^4}\\ &=\frac {a^3 (10 A b-7 a B) \sqrt {x} \sqrt {a+b x}}{128 b^4}-\frac {a^2 (10 A b-7 a B) x^{3/2} \sqrt {a+b x}}{192 b^3}+\frac {a (10 A b-7 a B) x^{5/2} \sqrt {a+b x}}{240 b^2}+\frac {(10 A b-7 a B) x^{7/2} \sqrt {a+b x}}{40 b}+\frac {B x^{7/2} (a+b x)^{3/2}}{5 b}-\frac {a^4 (10 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.32, size = 146, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {15 a^{7/2} (7 a B-10 A b) \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {\frac {b x}{a}+1}}+\sqrt {b} \sqrt {x} \left (-105 a^4 B+10 a^3 b (15 A+7 B x)-4 a^2 b^2 x (25 A+14 B x)+16 a b^3 x^2 (5 A+3 B x)+96 b^4 x^3 (5 A+4 B x)\right )\right )}{1920 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.30, size = 173, normalized size = 0.90 \begin {gather*} \frac {\left (10 a^4 A b-7 a^5 B\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{128 b^{9/2}}+\frac {\sqrt {a+b x} \left (-105 a^4 B \sqrt {x}+150 a^3 A b \sqrt {x}+70 a^3 b B x^{3/2}-100 a^2 A b^2 x^{3/2}-56 a^2 b^2 B x^{5/2}+80 a A b^3 x^{5/2}+48 a b^3 B x^{7/2}+480 A b^4 x^{7/2}+384 b^4 B x^{9/2}\right )}{1920 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.47, size = 295, normalized size = 1.54 \begin {gather*} \left [-\frac {15 \, {\left (7 \, 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} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{5}}, -\frac {15 \, {\left (7 \, 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} - 105 \, B a^{4} b + 150 \, A a^{3} b^{2} + 48 \, {\left (B a b^{4} + 10 \, A b^{5}\right )} x^{3} - 8 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} + 10 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{5}}\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}-96 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}-160 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}+112 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+150 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 )-105 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+200 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x -140 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -300 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+210 \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 {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.92, size = 242, normalized size = 1.26 \begin {gather*} \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B x^{2}}{5 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a x}{40 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{2} x}{32 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A x}{4 \, b} + \frac {7 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {9}{2}}} - \frac {5 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} - \frac {7 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a^{2}}{48 \, b^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{24 \, b^{2}} \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 x^{5/2}\,\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 [C] time = 48.22, size = 2370, normalized size = 12.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________