Optimal. Leaf size=151 \[ \frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {52, 65, 223,
212} \begin {gather*} -\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}}+\frac {\sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 b^2 d}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \sqrt {a+b x} (c+d x)^{3/2} \, dx &=\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}+\frac {(b c-a d) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{2 b}\\ &=\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}+\frac {(b c-a d)^2 \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 b^2}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^2 d}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \text {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^3 d}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \text {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^3 d}\\ &=\frac {(b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 b^2 d}+\frac {(b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 b^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{3 b}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 127, normalized size = 0.84 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 d^2+2 a b d (4 c+d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 b^2 d}-\frac {(b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{5/2} d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 173, normalized size = 1.15
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{2}}}{3 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 d}\) | \(173\) |
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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Fricas [A]
time = 0.93, size = 410, normalized size = 2.72 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} c^{3} - 3 \, 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} + 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (7 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b^{3} d^{2}}, \frac {3 \, {\left (b^{3} c^{3} - 3 \, 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} + 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} + 2 \, {\left (7 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b^{3} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 576 vs.
\(2 (119) = 238\).
time = 1.19, size = 576, normalized size = 3.81 \begin {gather*} -\frac {\frac {24 \, {\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 c {\left | b \right |}}{b^{2}} - \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}} + \frac {b^{6} c d^{3} - 13 \, a b^{5} d^{4}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{7} c^{2} d^{2} + 2 \, a b^{6} c d^{3} - 11 \, a^{2} b^{5} d^{4}\right )}}{b^{7} d^{4}}\right )} - \frac {3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 5 \, 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^{2}}\right )} d {\left | b \right |}}{b} - \frac {6 \, {\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 )} c {\left | b \right |}}{b^{2}} - \frac {6 \, {\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 )} a d {\left | b \right |}}{b^{3}}}{24 \, b} \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 \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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