Optimal. Leaf size=148 \[ -\frac {(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{5/2}}+\frac {\sqrt {a+b x} \sqrt {d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac {2 (a+b x)^{3/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {78, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac {(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{5/2}}-\frac {2 (a+b x)^{3/2} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(3 b B d-2 A b e-a B e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(3 b B d-2 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^2 (b d-a e)}-\frac {(3 b B d-2 A b e-a B e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{2 e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(3 b B d-2 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^2 (b d-a e)}-\frac {(3 b B d-2 A b e-a B e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(3 b B d-2 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^2 (b d-a e)}-\frac {(3 b B d-2 A b e-a B e) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{e (b d-a e) \sqrt {d+e x}}+\frac {(3 b B d-2 A b e-a B e) \sqrt {a+b x} \sqrt {d+e x}}{e^2 (b d-a e)}-\frac {(3 b B d-2 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.43, size = 129, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a+b x} (-2 A e+3 B d+B e x)}{e^2 \sqrt {d+e x}}-\frac {\sqrt {d+e x} (-a B e-2 A b e+3 b B d) \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{e^{5/2} \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.22, size = 151, normalized size = 1.02 \begin {gather*} \frac {(a B e+2 A b e-3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{5/2}}-\frac {\sqrt {a+b x} \left (\frac {2 A e^2 (a+b x)}{d+e x}-\frac {2 B d e (a+b x)}{d+e x}-a B e-2 A b e+3 b B d\right )}{e^2 \sqrt {d+e x} \left (\frac {e (a+b x)}{d+e x}-b\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.79, size = 358, normalized size = 2.42 \begin {gather*} \left [-\frac {{\left (3 \, B b d^{2} - {\left (B a + 2 \, A b\right )} d e + {\left (3 \, B b d e - {\left (B a + 2 \, A b\right )} e^{2}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (B b e^{2} x + 3 \, B b d e - 2 \, A b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{4 \, {\left (b e^{4} x + b d e^{3}\right )}}, \frac {{\left (3 \, B b d^{2} - {\left (B a + 2 \, A b\right )} d e + {\left (3 \, B b d e - {\left (B a + 2 \, A b\right )} e^{2}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (B b e^{2} x + 3 \, B b d e - 2 \, A b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b e^{4} x + b d e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.57, size = 154, normalized size = 1.04 \begin {gather*} \frac {{\left (3 \, B b d {\left | b \right |} - B a {\left | b \right |} e - 2 \, A b {\left | b \right |} e\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\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 |}\right )}{b^{\frac {3}{2}}} + \frac {{\left (\frac {{\left (b x + a\right )} B {\left | b \right |} e^{\left (-1\right )}}{b} + \frac {{\left (3 \, B b^{2} d {\left | b \right |} e - B a b {\left | b \right |} e^{2} - 2 \, A b^{2} {\left | b \right |} e^{2}\right )} e^{\left (-3\right )}}{b^{2}}\right )} \sqrt {b x + a}}{\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 386, normalized size = 2.61 \begin {gather*} \frac {\sqrt {b x +a}\, \left (2 A b \,e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+B a \,e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-3 B b d e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 A b d e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+B a d e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-3 B b \,d^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B e x -4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A e +6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B d \right )}{2 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e x +d}\, e^{2}} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________