Optimal. Leaf size=304 \[ -2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{96} \sqrt {a+b x} (c+d x)^{3/2} \left (\frac {3 a^2 d}{b}+50 a c-\frac {5 b c^2}{d}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2-55 a b^2 c^2 d+5 b^3 c^3\right )}{64 b^2 d}-\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 d} \]
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Rubi [A] time = 0.29, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-17 a^2 b c d^2+3 a^3 d^3-55 a b^2 c^2 d+5 b^3 c^3\right )}{64 b^2 d}-\frac {\left (-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}}+\frac {1}{96} \sqrt {a+b x} (c+d x)^{3/2} \left (\frac {3 a^2 d}{b}+50 a c-\frac {5 b c^2}{d}\right )-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 d} \end {gather*}
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)^{5/2}}{x} \, dx &=\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-4 a c+\frac {1}{2} (-5 b c-3 a d) x\right )}{x} \, dx\\ &=\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {(c+d x)^{3/2} \left (-12 a^2 c d+\frac {1}{4} \left (5 b^2 c^2-50 a b c d-3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 d}\\ &=\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {\sqrt {c+d x} \left (-24 a^2 b c^2 d+\frac {3}{8} \left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 b d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {-24 a^2 b^2 c^3 d+\frac {3}{16} \left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 b^2 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\left (a^2 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^2 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\left (2 a^2 c^3\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 (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\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 )}{64 b^3 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\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 )}{64 b^3 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}}\\ \end {align*}
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Mathematica [B] time = 2.93, size = 910, normalized size = 2.99 \begin {gather*} \frac {\sqrt {c+d x} \left (-15 b^4 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^4+15 b \sqrt {d} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^3+180 a b^3 d (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^3-384 a^{3/2} b d^{3/2} (b c-a d)^{3/2} \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) c^{5/2}+337 a d^{3/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+118 b d^{3/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c^2+270 a^2 b^2 d^2 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c^2+\frac {57 a^2 d^{5/2} (b c-a d)^{5/2} \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c}{b}+136 b d^{5/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c+244 a d^{5/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} c-60 a^3 b d^3 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) c+48 b d^{7/2} (b c-a d)^{5/2} x^3 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+72 a d^{7/2} (b c-a d)^{5/2} x^2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}+\frac {6 a^2 d^{7/2} (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}}{b}+9 a^4 d^4 (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )-9 a^3 d^{7/2} \sqrt {b c-a d} \sqrt {a+b x} (c+d x)^2 \sqrt {\frac {b (c+d x)}{b c-a d}}\right )}{192 d^{3/2} (b c-a d)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.76, size = 754, normalized size = 2.48 \begin {gather*} -2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {\left (3 a^4 d^4-20 a^3 b c d^3+90 a^2 b^2 c^2 d^2+60 a b^3 c^3 d-5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{5/2} d^{3/2}}+\frac {-\frac {9 a^4 b^3 d^4 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {33 a^4 b^2 d^5 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {9 a^4 d^7 \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {33 a^4 b d^6 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {60 a^3 b^4 c d^3 (c+d x)^{7/2}}{(a+b x)^{7/2}}-\frac {220 a^3 b^3 c d^4 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {92 a^3 b^2 c d^5 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {60 a^3 b c d^6 \sqrt {c+d x}}{\sqrt {a+b x}}-\frac {270 a^2 b^5 c^2 d^2 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {990 a^2 b^4 c^2 d^3 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {546 a^2 b^3 c^2 d^4 (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {114 a^2 b^2 c^2 d^5 \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {15 b^7 c^4 (c+d x)^{7/2}}{(a+b x)^{7/2}}+\frac {73 b^6 c^4 d (c+d x)^{5/2}}{(a+b x)^{5/2}}+\frac {204 a b^6 c^3 d (c+d x)^{7/2}}{(a+b x)^{7/2}}-\frac {55 b^5 c^4 d^2 (c+d x)^{3/2}}{(a+b x)^{3/2}}-\frac {876 a b^5 c^3 d^2 (c+d x)^{5/2}}{(a+b x)^{5/2}}+\frac {15 b^4 c^4 d^3 \sqrt {c+d x}}{\sqrt {a+b x}}+\frac {660 a b^4 c^3 d^3 (c+d x)^{3/2}}{(a+b x)^{3/2}}-\frac {180 a b^3 c^3 d^4 \sqrt {c+d x}}{\sqrt {a+b x}}}{192 b^2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 53.42, size = 1481, normalized size = 4.87
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 828, normalized size = 2.72 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (9 \sqrt {a c}\, a^{4} d^{4} \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 )-60 \sqrt {a c}\, a^{3} b c \,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 )-384 \sqrt {b d}\, a^{2} b^{2} c^{3} 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 )+270 \sqrt {a c}\, a^{2} b^{2} c^{2} 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 )+180 \sqrt {a c}\, a \,b^{3} c^{3} 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 )-15 \sqrt {a c}\, b^{4} c^{4} \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 )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} d^{3} x^{3}+144 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} d^{3} x^{2}+272 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c \,d^{2} x^{2}+12 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b \,d^{3} x +488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c \,d^{2} x +236 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{3} c^{2} d x -18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} d^{3}+114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b c \,d^{2}+674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{2} c^{2} d +30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{3} c^{3}\right )}{384 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d} \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x} \,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 {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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