Integrand size = 19, antiderivative size = 189 \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=-\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {3 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {52, 65, 223, 212} \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\frac {3 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^3}{64 b^2 d^2}+\frac {(a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^2}{32 b^2 d}+\frac {(a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b} \]
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Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {(3 (b c-a d)) \int (a+b x)^{3/2} \sqrt {c+d x} \, dx}{8 b} \\ & = \frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {(b c-a d)^2 \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{16 b^2} \\ & = \frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}-\frac {\left (3 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{64 b^2 d} \\ & = -\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^2 d^2} \\ & = -\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^3 d^2} \\ & = -\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {\left (3 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^3 d^2} \\ & = -\frac {3 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d^2}+\frac {(b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{32 b^2 d}+\frac {(b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{8 b^2}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}+\frac {3 (b c-a d)^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.88 \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^3 d^3+a^2 b d^2 (11 c+2 d x)+a b^2 d \left (11 c^2+44 c d x+24 d^2 x^2\right )+b^3 \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )}{64 b^2 d^2}+\frac {3 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{5/2} d^{5/2}} \]
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Time = 0.53 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {5}{2}}}{4 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {5}{2}}}{3 d}-\frac {\left (-a d +b c \right ) \left (\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {d x +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {b d}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {b d}}\right )}{4 b}\right )}{6 d}\right )}{8 d}\) | \(206\) |
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Time = 0.25 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.83 \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\left [\frac {3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (16 \, b^{4} d^{4} x^{3} - 3 \, b^{4} c^{3} d + 11 \, a b^{3} c^{2} d^{2} + 11 \, a^{2} b^{2} c d^{3} - 3 \, a^{3} b d^{4} + 24 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{256 \, b^{3} d^{3}}, -\frac {3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (16 \, b^{4} d^{4} x^{3} - 3 \, b^{4} c^{3} d + 11 \, a b^{3} c^{2} d^{2} + 11 \, a^{2} b^{2} c d^{3} - 3 \, a^{3} b d^{4} + 24 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{2} + 22 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{128 \, b^{3} d^{3}}\right ] \]
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\[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\int \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (151) = 302\).
Time = 0.43 (sec) , antiderivative size = 1071, normalized size of antiderivative = 5.67 \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b x)^{3/2} (c+d x)^{3/2} \, dx=\int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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