Optimal. Leaf size=111 \[ -\frac {2 (a+b x)^{3/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}+\frac {2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{5/2}}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.06, antiderivative size = 111, 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, 47, 63, 217, 206} \[ -\frac {2 (a+b x)^{3/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{5/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {B \int \frac {\sqrt {a+b x}}{(d+e x)^{3/2}} \, dx}{e}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}}+\frac {(b B) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{e^2}\\ &=-\frac {2 (B d-A e) (a+b x)^{3/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 B \sqrt {a+b x}}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 148, normalized size = 1.33 \[ \frac {2 \left (\frac {\sqrt {e} \sqrt {a+b x} \left (a e (A e+2 B d+3 B e x)+A b e^2 x-b B d (3 d+4 e x)\right )}{b d-a e}+\frac {3 B (b d-a e)^{3/2} \left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{b}\right )}{3 e^{5/2} (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.17, size = 515, normalized size = 4.64 \[ \left [\frac {3 \, {\left (B b d^{3} - B a d^{2} e + {\left (B b d e^{2} - B a e^{3}\right )} x^{2} + 2 \, {\left (B b d^{2} e - B a d e^{2}\right )} x\right )} \sqrt {\frac {b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - 4 \, {\left (3 \, B b d^{2} - 2 \, B a d e - A a e^{2} + {\left (4 \, B b d e - {\left (3 \, B a + A b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, {\left (b d^{3} e^{2} - a d^{2} e^{3} + {\left (b d e^{4} - a e^{5}\right )} x^{2} + 2 \, {\left (b d^{2} e^{3} - a d e^{4}\right )} x\right )}}, -\frac {3 \, {\left (B b d^{3} - B a d^{2} e + {\left (B b d e^{2} - B a e^{3}\right )} x^{2} + 2 \, {\left (B b d^{2} e - B a d e^{2}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) + 2 \, {\left (3 \, B b d^{2} - 2 \, B a d e - A a e^{2} + {\left (4 \, B b d e - {\left (3 \, B a + A b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (b d^{3} e^{2} - a d^{2} e^{3} + {\left (b d e^{4} - a e^{5}\right )} x^{2} + 2 \, {\left (b d^{2} e^{3} - a d e^{4}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.81, size = 198, normalized size = 1.78 \[ -\frac {2 \, B {\left | b \right |} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (4 \, B b^{4} d {\left | b \right |} e^{2} - 3 \, B a b^{3} {\left | b \right |} e^{3} - A b^{4} {\left | b \right |} e^{3}\right )} {\left (b x + a\right )}}{b^{3} d e^{3} - a b^{2} e^{4}} + \frac {3 \, {\left (B b^{5} d^{2} {\left | b \right |} e - 2 \, B a b^{4} d {\left | b \right |} e^{2} + B a^{2} b^{3} {\left | b \right |} e^{3}\right )}}{b^{3} d e^{3} - a b^{2} e^{4}}\right )}}{3 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 503, normalized size = 4.53 \[ -\frac {\left (-3 B a b \,e^{3} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+3 B \,b^{2} d \,e^{2} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-6 B a b d \,e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+6 B \,b^{2} d^{2} e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-3 B a b \,d^{2} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+3 B \,b^{2} d^{3} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A b \,e^{2} x +6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a \,e^{2} x -8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B b d e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A a \,e^{2}+4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a d e -6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B b \,d^{2}\right ) \sqrt {b x +a}}{3 \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \left (e x +d \right )^{\frac {3}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \sqrt {a + b x}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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