Optimal. Leaf size=199 \[ \frac {3 (b d-a e) (-5 a B e+4 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{7/2} \sqrt {e}}+\frac {3 \sqrt {a+b x} \sqrt {d+e x} (-5 a B e+4 A b e+b B d)}{4 b^3}+\frac {\sqrt {a+b x} (d+e x)^{3/2} (-5 a B e+4 A b e+b B d)}{2 b^2 (b d-a e)}-\frac {2 (d+e x)^{5/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {\sqrt {a+b x} (d+e x)^{3/2} (-5 a B e+4 A b e+b B d)}{2 b^2 (b d-a e)}+\frac {3 \sqrt {a+b x} \sqrt {d+e x} (-5 a B e+4 A b e+b B d)}{4 b^3}+\frac {3 (b d-a e) (-5 a B e+4 A b e+b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{7/2} \sqrt {e}}-\frac {2 (d+e x)^{5/2} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{(a+b x)^{3/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(b B d+4 A b e-5 a B e) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}} \, dx}{b (b d-a e)}\\ &=\frac {(b B d+4 A b e-5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(3 (b B d+4 A b e-5 a B e)) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{4 b^2}\\ &=\frac {3 (b B d+4 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^3}+\frac {(b B d+4 A b e-5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(3 (b d-a e) (b B d+4 A b e-5 a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 b^3}\\ &=\frac {3 (b B d+4 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^3}+\frac {(b B d+4 A b e-5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(3 (b d-a e) (b B d+4 A b e-5 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 )}{4 b^4}\\ &=\frac {3 (b B d+4 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^3}+\frac {(b B d+4 A b e-5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {(3 (b d-a e) (b B d+4 A b e-5 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 )}{4 b^4}\\ &=\frac {3 (b B d+4 A b e-5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^3}+\frac {(b B d+4 A b e-5 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{2 b^2 (b d-a e)}-\frac {2 (A b-a B) (d+e x)^{5/2}}{b (b d-a e) \sqrt {a+b x}}+\frac {3 (b d-a e) (b B d+4 A b e-5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{7/2} \sqrt {e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.60, size = 161, normalized size = 0.81 \[ \frac {\sqrt {d+e x} \left (\frac {B \left (-15 a^2 e+a b (13 d-5 e x)+b^2 x (5 d+2 e x)\right )+4 A b (3 a e-2 b d+b e x)}{\sqrt {a+b x}}+\frac {3 \sqrt {b d-a e} (-5 a B e+4 A b e+b B d) \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{\sqrt {e} \sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{4 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.14, size = 584, normalized size = 2.93 \[ \left [\frac {3 \, {\left (B a b^{2} d^{2} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e + {\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + {\left (B b^{3} d^{2} - 2 \, {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, B b^{3} e^{2} x^{2} + {\left (13 \, B a b^{2} - 8 \, A b^{3}\right )} d e - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2} + {\left (5 \, B b^{3} d e - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{16 \, {\left (b^{5} e x + a b^{4} e\right )}}, -\frac {3 \, {\left (B a b^{2} d^{2} - 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e + {\left (5 \, B a^{3} - 4 \, A a^{2} b\right )} e^{2} + {\left (B b^{3} d^{2} - 2 \, {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} d e + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, B b^{3} e^{2} x^{2} + {\left (13 \, B a b^{2} - 8 \, A b^{3}\right )} d e - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2} + {\left (5 \, B b^{3} d e - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{8 \, {\left (b^{5} e x + a b^{4} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.98, size = 337, normalized size = 1.69 \[ \frac {1}{4} \, \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} B {\left | b \right |} e}{b^{5}} + \frac {{\left (5 \, B b^{10} d {\left | b \right |} e^{2} - 9 \, B a b^{9} {\left | b \right |} e^{3} + 4 \, A b^{10} {\left | b \right |} e^{3}\right )} e^{\left (-2\right )}}{b^{14}}\right )} - \frac {3 \, {\left (B b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} - 6 \, B a b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 4 \, A b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 5 \, B a^{2} \sqrt {b} {\left | b \right |} e^{\frac {5}{2}} - 4 \, A a b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{2}}\right )} e^{\left (-1\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{8 \, b^{5}} + \frac {4 \, {\left (B a b^{\frac {5}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} - A b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} - 2 \, B a^{2} b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 2 \, A a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + B a^{3} \sqrt {b} {\left | b \right |} e^{\frac {5}{2}} - A a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{2}}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 740, normalized size = 3.72 \[ -\frac {\sqrt {e x +d}\, \left (12 A a \,b^{2} 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 )-12 A \,b^{3} 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 )-15 B \,a^{2} 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 )+18 B a \,b^{2} 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 )-3 B \,b^{3} d^{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 )+12 A \,a^{2} b \,e^{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 )-12 A a \,b^{2} 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 )-15 B \,a^{3} e^{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 )+18 B \,a^{2} 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 )-3 B a \,b^{2} 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 )-4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,b^{2} e \,x^{2}-8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A \,b^{2} e x +10 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a b e x -10 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,b^{2} d x -24 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A a b e +16 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A \,b^{2} d +30 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,a^{2} e -26 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a b d \right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, \sqrt {b x +a}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________