Optimal. Leaf size=125 \[ -\frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d} \]
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Rubi [A] time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {80, 50, 63, 217, 206} \[ -\frac {(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (3 a d+b c)}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x \sqrt {c+d x}}{\sqrt {a+b x}} \, dx &=\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d}-\frac {(b c+3 a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{4 b d}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d}-\frac {((b c-a d) (b c+3 a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^2 d}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d}-\frac {((b c-a d) (b c+3 a d)) \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^3 d}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d}-\frac {((b c-a d) (b c+3 a d)) \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^3 d}\\ &=-\frac {(b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 b^2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b d}-\frac {(b c-a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 121, normalized size = 0.97 \[ \frac {\sqrt {c+d x} \left (\sqrt {d} \sqrt {a+b x} (b (c+2 d x)-3 a d)-\frac {\sqrt {b c-a d} (3 a d+b c) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{4 b^2 d^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 304, normalized size = 2.43 \[ \left [-\frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b^{3} d^{2}}, \frac {{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x + b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b^{3} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 132, normalized size = 1.06 \[ \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, b x + 2 \, a + \frac {b c d - 5 \, a d^{2}}{d^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} c^{2} + 2 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d}\right )} {\left | b \right |}}{4 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 251, normalized size = 2.01 \[ \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 a^{2} d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-2 a b c d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-b^{2} c^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d +2 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{2} d} \]
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 [B] time = 18.40, size = 584, normalized size = 4.67 \[ \frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-\frac {3\,a^2\,b\,d^2}{2}+a\,b^2\,c\,d+\frac {b^3\,c^2}{2}\right )}{d^5\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {11\,a^2\,d^2}{2}+23\,a\,b\,c\,d+\frac {7\,b^2\,c^2}{2}\right )}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {8\,\sqrt {a}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (-\frac {3\,a^2\,d^2}{2}+a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{b^2\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {11\,a^2\,d^2}{2}+23\,a\,b\,c\,d+\frac {7\,b^2\,c^2}{2}\right )}{b\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}-\frac {\sqrt {a}\,\sqrt {c}\,\left (32\,a\,d+16\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {8\,\sqrt {a}\,b^2\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}+\frac {b^4}{d^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}}+\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+b\,c\right )}{2\,b^{5/2}\,d^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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