Optimal. Leaf size=116 \[ -\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b} \]
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Rubi [A] time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {50, 63, 217, 206} \begin {gather*} -\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \sqrt {a+b x} \sqrt {c+d x} \, dx &=\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}+\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 b}\\ &=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b d}\\ &=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 b^2 d}\\ &=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 b^2 d}\\ &=\frac {(b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b d}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 b}-\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 118, normalized size = 1.02 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) (a d+b (c+2 d x))-(b c-a d)^{5/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 b^2 d^{3/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 129, normalized size = 1.11 \begin {gather*} \frac {\sqrt {c+d x} (b c-a d)^2 \left (\frac {b (c+d x)}{a+b x}+d\right )}{4 b d \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^2}-\frac {(b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{3/2} d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 300, normalized size = 2.59 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{2} d^{2}}, \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{2} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.49, size = 232, normalized size = 2.00 \begin {gather*} -\frac {\frac {4 \, {\left (\frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d}} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a}\right )} a {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} {\left | b \right |}}{b^{2}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 305, normalized size = 2.63 \begin {gather*} -\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{8 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, b}+\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{4 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}-\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b \,c^{2} \ln \left (\frac {b d x +\frac {1}{2} a d +\frac {1}{2} b c}{\sqrt {b d}}+\sqrt {b d \,x^{2}+a c +\left (a d +b c \right ) x}\right )}{8 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}\, d}+\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, a}{4 b}-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, c}{4 d}+\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}}}{2 d} \end {gather*}
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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mupad [B] time = 0.12, size = 88, normalized size = 0.76 \begin {gather*} \left (\frac {x}{2}+\frac {a\,d+b\,c}{4\,b\,d}\right )\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}-\frac {\ln \left (a\,d+b\,c+2\,b\,d\,x+2\,\sqrt {b}\,\sqrt {d}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}\right )\,{\left (a\,d-b\,c\right )}^2}{8\,b^{3/2}\,d^{3/2}} \end {gather*}
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 \begin {gather*} \int \sqrt {a + b x} \sqrt {c + d x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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