∫ √ ____ a + bx(A+Bx)______________

Optimal. Leaf size=140 \[ -\frac {(3 b B d-4 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {(b d-a e) (3 b B d-4 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{3/2} e^{5/2}} \]

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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) (a B e-4 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{3/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (a B e-4 A b e+3 b B d)}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e} \end {gather*}

\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{\sqrt {d+e x}} \, dx &=\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {\left (2 A b e-B \left (\frac {3 b d}{2}+\frac {a e}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{2 b e}\\ &=-\frac {(3 b B d-4 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {((b d-a e) (3 b B d-4 A b e+a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 b e^2}\\ &=-\frac {(3 b B d-4 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {((b d-a e) (3 b B d-4 A b e+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^2 e^2}\\ &=-\frac {(3 b B d-4 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {((b d-a e) (3 b B d-4 A b e+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^2 e^2}\\ &=-\frac {(3 b B d-4 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b e^2}+\frac {B (a+b x)^{3/2} \sqrt {d+e x}}{2 b e}+\frac {(b d-a e) (3 b B d-4 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{3/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 117, normalized size = 0.84 \begin {gather*} \frac {b \sqrt {a+b x} \sqrt {d+e x} (a B e+b (-3 B d+4 A e+2 B e x))+\sqrt {\frac {b}{e}} (b d-a e) (-3 b B d+4 A b e-a B e) \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{e}} \sqrt {d+e x}\right )}{4 b^2 e^2} \end {gather*}

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(114)=228\).
time = 0.09, size = 376, normalized size = 2.69


method	result	size

default	\(\frac {\sqrt {b x +a}\, \sqrt {e x +d}\, \left (4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,e^{2}-4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d e -B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{2}-2 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d e +3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2}+4 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b e x +8 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b e -6 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b d +2 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a e \right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, e^{2} b \sqrt {b e}}\)	\(376\)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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Fricas [A]
time = 1.64, size = 362, normalized size = 2.59 \begin {gather*} \left [\frac {{\left ({\left (3 \, B b^{2} d^{2} - 2 \, {\left (B a b + 2 \, A b^{2}\right )} d e - {\left (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 (3 \, B b^{2} d e - {\left (2 \, B b^{2} x + B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{16 \, b^{2}}, -\frac {{\left ({\left (3 \, B b^{2} d^{2} - 2 \, {\left (B a b + 2 \, A b^{2}\right )} d e - {\left (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 (3 \, B b^{2} d e - {\left (2 \, B b^{2} x + B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{8 \, b^{2}}\right ] \end {gather*}

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 57645 vs. \(2 (133) = 266\).
time = 143.25, size = 57645, normalized size = 411.75 \begin {gather*} \text {Too large to display} \end {gather*}

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Giac [A]
time = 1.02, size = 175, normalized size = 1.25 \begin {gather*} \frac {{\left (\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 e^{\left (-1\right )}}{b^{2}} - \frac {{\left (3 \, B b^{3} d e + B a b^{2} e^{2} - 4 \, A b^{3} e^{2}\right )} e^{\left (-3\right )}}{b^{4}}\right )} - \frac {{\left (3 \, B b^{2} d^{2} - 2 \, B a b d e - 4 \, A b^{2} d e - B a^{2} e^{2} + 4 \, A a b e^{2}\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}}}\right )} b}{4 \, {\left | b \right |}} \end {gather*}

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Mupad [B]
time = 22.84, size = 872, normalized size = 6.23 \begin {gather*} \frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {B\,a^2\,b^2\,e^2}{2}+B\,a\,b^3\,d\,e-\frac {3\,B\,b^4\,d^2}{2}\right )}{e^6\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {7\,B\,a^2\,b\,e^2}{2}+23\,B\,a\,b^2\,d\,e+\frac {11\,B\,b^3\,d^2}{2}\right )}{e^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {7\,B\,a^2\,e^2}{2}+23\,B\,a\,b\,d\,e+\frac {11\,B\,b^2\,d^2}{2}\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (\frac {B\,a^2\,e^2}{2}+B\,a\,b\,d\,e-\frac {3\,B\,b^2\,d^2}{2}\right )}{b\,e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}-\frac {\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (32\,B\,d\,b^2+16\,B\,a\,e\,b\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {8\,B\,a^{3/2}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}-\frac {8\,B\,a^{3/2}\,b^2\,\sqrt {d}\,{\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 (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (2\,A\,d\,b^2+2\,A\,a\,e\,b\right )}{e^3\,\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}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}-\frac {8\,A\,\sqrt {a}\,b\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^2\,{\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 )}{\sqrt {b}\,e^{3/2}}-\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 (a\,e+3\,b\,d\right )}{2\,b^{3/2}\,e^{5/2}} \end {gather*}

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3.23.3 \(\int \frac {\sqrt {a+b x} (A+B x)}{\sqrt {d+e x}} \, dx\) [2203]