Optimal. Leaf size=171 \[ -\frac {(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{7/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) (a d+5 b c)}{8 b d^3}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+5 b c)}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d} \]
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Rubi [A] time = 0.10, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{7/2}}-\frac {(a+b x)^{3/2} \sqrt {c+d x} (a d+5 b c)}{12 b d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d) (a d+5 b c)}{8 b d^3}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d} \end {gather*}
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 (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx &=\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}-\frac {(5 b c+a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 b d}\\ &=-\frac {(5 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}+\frac {((b c-a d) (5 b c+a d)) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b d^2}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^3}-\frac {(5 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d^3}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^3}-\frac {(5 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d)^2 (5 b c+a d)\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 )}{8 b^2 d^3}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^3}-\frac {(5 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}-\frac {\left ((b c-a d)^2 (5 b c+a d)\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 )}{8 b^2 d^3}\\ &=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{8 b d^3}-\frac {(5 b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{12 b d^2}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b d}-\frac {(b c-a d)^2 (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 159, normalized size = 0.93 \begin {gather*} \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (3 a^2 d^2+2 a b d (7 d x-11 c)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )-3 (b c-a d)^{5/2} (a d+5 b c) \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 )}{24 b^2 d^{7/2} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 214, normalized size = 1.25 \begin {gather*} \frac {\sqrt {c+d x} (b c-a d)^2 \left (\frac {15 b^3 c (c+d x)^2}{(a+b x)^2}+\frac {3 a b^2 d (c+d x)^2}{(a+b x)^2}-\frac {40 b^2 c d (c+d x)}{a+b x}-\frac {8 a b d^2 (c+d x)}{a+b x}-3 a d^3+33 b c d^2\right )}{24 b d^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{a+b x}-d\right )^3}-\frac {(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 412, normalized size = 2.41 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{2} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{2} d^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.49, size = 213, normalized size = 1.25 \begin {gather*} \frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2} d} - \frac {5 \, b^{3} c d^{3} + a b^{2} d^{4}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\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^{3}}\right )} b}{24 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 395, normalized size = 2.31 \begin {gather*} -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 a^{3} d^{3} \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 )+9 a^{2} b c \,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 )-27 a \,b^{2} c^{2} 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 )+15 b^{3} c^{3} \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 )-16 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} d^{2} x^{2}-28 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x +20 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}+44 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d -30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,d^{3}} \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\,{\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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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