Optimal. Leaf size=113 \[ \frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 113, 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 {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^2}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx &=\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {(3 (b c-a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \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 d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {\left (3 (b c-a d)^2\right ) \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 d^2}\\ &=-\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}+\frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 119, normalized size = 1.05 \begin {gather*} \frac {\sqrt {d} \sqrt {a+b x} (c+d x) (5 a d-3 b c+2 b d x)+\frac {3 (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 )}{b}}{4 d^{5/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 134, normalized size = 1.19 \begin {gather*} \frac {3 (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 \sqrt {b} d^{5/2}}+\frac {(a d-b c)^2 \left (\frac {5 d \sqrt {c+d x}}{\sqrt {a+b x}}-\frac {3 b (c+d x)^{3/2}}{(a+b x)^{3/2}}\right )}{4 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 306, normalized size = 2.71 \begin {gather*} \left [\frac {3 \, {\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 - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, b d^{3}}, -\frac {3 \, {\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 - 3 \, b^{2} c d + 5 \, a b d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, b d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 139, normalized size = 1.23 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )}}{b d} - \frac {3 \, {\left (b c d - a d^{2}\right )}}{b d^{3}}\right )} - \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} 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^{2}}\right )} b}{4 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 308, normalized size = 2.73 \begin {gather*} \frac {3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{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 {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}}-\frac {3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b 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 {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d}+\frac {3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} 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 {b x +a}\, \sqrt {d x +c}\, \sqrt {b d}\, d^{2}}+\frac {3 \sqrt {b x +a}\, \sqrt {d x +c}\, a}{4 d}-\frac {3 \sqrt {b x +a}\, \sqrt {d x +c}\, b c}{4 d^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}{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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}}{\sqrt {c+d\,x}} \,d x \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 \frac {\left (a + b x\right )^{\frac {3}{2}}}{\sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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