Optimal. Leaf size=213 \[ -2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac {1}{8} \sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+8 a c-\frac {b c^2}{d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]
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Rubi [A] time = 0.20, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ -\frac {(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}}+\frac {1}{8} \sqrt {a+b x} \sqrt {c+d x} \left (\frac {a^2 d}{b}+8 a c-\frac {b c^2}{d}\right )-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 101
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx &=\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {1}{3} \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a c-\frac {3}{2} (b c+a d) x\right )}{x} \, dx\\ &=\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {\int \frac {\sqrt {c+d x} \left (-6 a^2 c d+\frac {3}{4} \left (b^2 c^2-8 a b c d-a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {\int \frac {-6 a^2 b c^2 d+\frac {3}{8} (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (a^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b d}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}+\left (2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\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}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left ((b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right )\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}\\ &=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 243, normalized size = 1.14 \[ \frac {\sqrt {d} \left (\sqrt {a+b x} (c+d x) \left (3 a^2 d^2+2 a b d (19 c+7 d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )-48 a^{3/2} b c^{3/2} d \sqrt {c+d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )-\frac {3 \sqrt {b c-a d} \left (a^3 d^3-9 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{b}}{24 b d^{3/2} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 13.29, size = 1193, normalized size = 5.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 587, normalized size = 2.76 \[ -\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \sqrt {a c}\, a^{3} d^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+48 \sqrt {b d}\, a^{2} b \,c^{2} d \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-27 \sqrt {a c}\, a^{2} b c \,d^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-27 \sqrt {a c}\, a \,b^{2} c^{2} d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 \sqrt {a c}\, b^{3} c^{3} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-16 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d^{2} x^{2}-28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a b \,d^{2} x -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} d^{2}-76 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a b c d -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{2} c^{2}\right )}{48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b d} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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