Optimal. Leaf size=111 \[ -\frac {2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}}-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}} \]
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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} \[ -\frac {2 (d+e x)^{3/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^{5/2}} \, dx &=-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {B \int \frac {\sqrt {d+e x}}{(a+b x)^{3/2}} \, dx}{b}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(B e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b^2}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 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^3}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(2 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^3}\\ &=-\frac {2 B \sqrt {d+e x}}{b^2 \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{3/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {2 B \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 114, normalized size = 1.03 \[ \frac {2 \sqrt {d+e x} \left ((d+e x) (B d-A e)-\frac {B (b d-a e)^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {e (a+b x)}{a e-b d}\right )}{b^2 \sqrt {\frac {b (d+e x)}{b d-a e}}}\right )}{3 e (a+b x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.83, size = 525, normalized size = 4.73 \[ \left [\frac {3 \, {\left (B a^{2} b d - B a^{3} e + {\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \, {\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt {\frac {e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {e}{b}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (3 \, B a^{2} e - {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} d - {\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{6 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + {\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \, {\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}, -\frac {3 \, {\left (B a^{2} b d - B a^{3} e + {\left (B b^{3} d - B a b^{2} e\right )} x^{2} + 2 \, {\left (B a b^{2} d - B a^{2} b e\right )} x\right )} \sqrt {-\frac {e}{b}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {e}{b}}}{2 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )}}\right ) - 2 \, {\left (3 \, B a^{2} e - {\left (2 \, B a b + A b^{2}\right )} d - {\left (3 \, B b^{2} d - {\left (4 \, B a b - A b^{2}\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + {\left (b^{5} d - a b^{4} e\right )} x^{2} + 2 \, {\left (a b^{4} d - a^{2} b^{3} e\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.81, size = 522, normalized size = 4.70 \[ -\frac {B {\left | b \right |} e^{\frac {1}{2}} \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 )}{b^{\frac {7}{2}}} - \frac {4 \, {\left (3 \, B b^{\frac {11}{2}} d^{3} {\left | b \right |} e^{\frac {1}{2}} - 10 \, B a b^{\frac {9}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} + A b^{\frac {11}{2}} d^{2} {\left | b \right |} e^{\frac {3}{2}} - 6 \, {\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} B b^{\frac {7}{2}} d^{2} {\left | b \right |} e^{\frac {1}{2}} + 11 \, B a^{2} b^{\frac {7}{2}} d {\left | b \right |} e^{\frac {5}{2}} - 2 \, A a b^{\frac {9}{2}} d {\left | b \right |} e^{\frac {5}{2}} + 12 \, {\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} B a b^{\frac {5}{2}} d {\left | b \right |} e^{\frac {3}{2}} + 3 \, {\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 )}^{4} B b^{\frac {3}{2}} d {\left | b \right |} e^{\frac {1}{2}} - 4 \, B a^{3} b^{\frac {5}{2}} {\left | b \right |} e^{\frac {7}{2}} + A a^{2} b^{\frac {7}{2}} {\left | b \right |} e^{\frac {7}{2}} - 6 \, {\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} B a^{2} b^{\frac {3}{2}} {\left | b \right |} e^{\frac {5}{2}} - 6 \, {\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 )}^{4} B a \sqrt {b} {\left | b \right |} e^{\frac {3}{2}} + 3 \, {\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 )}^{4} A b^{\frac {3}{2}} {\left | b \right |} e^{\frac {3}{2}}\right )}}{3 \, {\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 )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 503, normalized size = 4.53 \[ \frac {\left (3 B a \,b^{2} e^{2} x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-3 B \,b^{3} d e \,x^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+6 B \,a^{2} b \,e^{2} x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-6 B a \,b^{2} d e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+3 B \,a^{3} e^{2} \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-3 B \,a^{2} b d e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A \,b^{2} e x -8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a b e x +6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,b^{2} d x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A \,b^{2} d -6 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B \,a^{2} e +4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a b d \right ) \sqrt {e x +d}}{3 \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \left (b x +a \right )^{\frac {3}{2}} b^{2}} \]
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 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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