Optimal. Leaf size=140 \[ -\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(b d-a e) (b B d-4 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{5/2} e^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 140, 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 = {81, 52, 65, 223,
212} \begin {gather*} -\frac {(b d-a e) (3 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^{5/2} e^{3/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx &=\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}+\frac {\left (2 A b e-B \left (\frac {b d}{2}+\frac {3 a e}{2}\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}} \, dx}{2 b e}\\ &=-\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {((b d-a e) (b B d-4 A b e+3 a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 b^2 e}\\ &=-\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {((b d-a e) (b B d-4 A b e+3 a B e)) \text {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^3 e}\\ &=-\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {((b d-a e) (b B d-4 A b e+3 a B e)) \text {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^3 e}\\ &=-\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(b d-a e) (b B d-4 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{5/2} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 117, normalized size = 0.84 \begin {gather*} \frac {\frac {b^2 \sqrt {a+b x} \sqrt {d+e x} (4 A b e-3 a B e+b B (d+2 e x))}{e}+\left (\frac {b}{e}\right )^{3/2} (b d-a e) (b B d-4 A b e+3 a B e) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{e}} \sqrt {d+e x}\right )}{4 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs.
\(2(114)=228\).
time = 0.09, size = 375, normalized size = 2.68
method | result | size |
default | \(-\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,e^{2}-4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d e -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{2}+2 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d e +B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2}-4 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b e x -8 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b e +6 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a e -2 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b d \right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} e \sqrt {b e}}\) | \(375\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.18, size = 360, normalized size = 2.57 \begin {gather*} \left [\frac {{\left ({\left (B b^{2} d^{2} + 2 \, {\left (B a b - 2 \, A b^{2}\right )} d e - {\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} - 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) + 4 \, {\left (B b^{2} d e + {\left (2 \, B b^{2} x - 3 \, B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-2\right )}}{16 \, b^{3}}, \frac {{\left ({\left (B b^{2} d^{2} + 2 \, {\left (B a b - 2 \, A b^{2}\right )} d e - {\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (B b^{2} d e + {\left (2 \, B b^{2} x - 3 \, B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-2\right )}}{8 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs.
\(2 (118) = 236\).
time = 1.17, size = 238, normalized size = 1.70 \begin {gather*} -\frac {\frac {4 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{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}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A {\left | b \right |}}{b^{2}} - \frac {{\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{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}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} B {\left | b \right |}}{b^{3}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 22.14, size = 866, normalized size = 6.19 \begin {gather*} \frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {11\,B\,a^2\,e^2}{2}+23\,B\,a\,b\,d\,e+\frac {7\,B\,b^2\,d^2}{2}\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-\frac {3\,B\,a^2\,b\,e^2}{2}+B\,a\,b^2\,d\,e+\frac {B\,b^3\,d^2}{2}\right )}{e^5\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (-\frac {3\,B\,a^2\,e^2}{2}+B\,a\,b\,d\,e+\frac {B\,b^2\,d^2}{2}\right )}{b^2\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {11\,B\,a^2\,e^2}{2}+23\,B\,a\,b\,d\,e+\frac {7\,B\,b^2\,d^2}{2}\right )}{b\,e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}-\frac {\sqrt {a}\,\sqrt {d}\,\left (32\,B\,a\,e+16\,B\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {8\,B\,\sqrt {a}\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}-\frac {8\,B\,\sqrt {a}\,b^2\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {b^4}{e^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}}+\frac {\frac {\left (2\,A\,a\,e+2\,A\,b\,d\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{e^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}+\frac {\left (2\,A\,a\,e+2\,A\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}-\frac {8\,A\,\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {b^2}{e^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )}{b^{3/2}\,\sqrt {e}}+\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )\,\left (3\,a\,e+b\,d\right )}{2\,b^{5/2}\,e^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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