Optimal. Leaf size=127 \[ -\frac {(3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (3 b c-a d)}{d^2 (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.07, antiderivative size = 127, 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 = {78, 50, 63, 217, 206} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} (3 b c-a d)}{d^2 (b c-a d)}-\frac {(3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx &=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d (b c-a d)}\\ &=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-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 )}{b d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-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 )}{b d^2}\\ &=-\frac {2 c (a+b x)^{3/2}}{d (b c-a d) \sqrt {c+d x}}+\frac {(3 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{d^2 (b c-a d)}-\frac {(3 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 126, normalized size = 0.99 \begin {gather*} \frac {\sqrt {d} \sqrt {a+b x} (3 c+d x)-\frac {\left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \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 \sqrt {b c-a d}}}{d^{5/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 116, normalized size = 0.91 \begin {gather*} \frac {(a d-3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}}+\frac {\sqrt {a+b x} \left (\frac {2 c d (a+b x)}{c+d x}+a d-3 b c\right )}{d^2 \sqrt {c+d x} \left (\frac {d (a+b x)}{c+d x}-b\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 308, normalized size = 2.43 \begin {gather*} \left [-\frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\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 (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b d^{4} x + b c d^{3}\right )}}, \frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\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 (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b d^{4} x + b c d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 134, normalized size = 1.06 \begin {gather*} \frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} {\left | b \right |}}{b d} + \frac {3 \, b^{2} c d {\left | b \right |} - a b d^{2} {\left | b \right |}}{b^{2} d^{3}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c {\left | b \right |} - a d {\left | b \right |}\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} b d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 264, normalized size = 2.08 \begin {gather*} \frac {\sqrt {b x +a}\, \left (a \,d^{2} x \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 )-3 b c d x \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 )+a 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 )-3 b \,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 )+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, d x +6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, c \right )}{2 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {d x +c}\, d^{2}} \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 {x\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{3/2}} \,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 {x \sqrt {a + b x}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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