Optimal. Leaf size=319 \[ \frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )}{12 c}+\frac {5}{8} \sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac {5 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{12 c}-\frac {5}{4} \sqrt {a} \sqrt {c} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \]
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Rubi [A] time = 0.39, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {97, 149, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )}{12 c}+\frac {5}{8} \sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac {5 (a d+b c) \left (a^2 d^2+14 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 \sqrt {b} \sqrt {d}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{12 c}-\frac {5}{4} \sqrt {a} \sqrt {c} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{2} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x^2} \, dx\\ &=-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {5}{4} (3 b c+a d) (b c+3 a d)+\frac {5}{2} b d (5 b c+3 a d) x\right )}{x} \, dx}{2 c}\\ &=\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {15}{4} a d (3 b c+a d) (b c+3 a d)+5 b d \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 c d}\\ &=\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {15}{2} a b c d (3 b c+a d) (b c+3 a d)+\frac {15}{2} b^2 c d \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 b c d}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\frac {15}{2} a b^2 c^2 d (3 b c+a d) (b c+3 a d)+\frac {15}{4} b^2 c d (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 b^2 c d}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{8} (5 a c (3 b c+a d) (b c+3 a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{16} \left (5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{4} (5 a c (3 b c+a d) (b c+3 a d)) \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 c+a d) \left (b^2 c^2+14 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}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 (b c+a d) \left (b^2 c^2+14 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}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {5 (b c+a d) \left (b^2 c^2+14 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 \sqrt {b} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 2.79, size = 282, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 \left (4 c^2+18 c d x-11 d^2 x^2\right )+2 a b x \left (-27 c^2+61 c d x+13 d^2 x^2\right )+b^2 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )}{24 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} \left (3 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {5 \sqrt {c+d x} \left (a^3 d^3+15 a^2 b c d^2+15 a b^2 c^2 d+b^3 c^3\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{8 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.88, size = 839, normalized size = 2.63 \begin {gather*} \frac {-\frac {15 b^2 d^3 (c+d x)^{9/2} a^5}{(a+b x)^{9/2}}+\frac {40 b d^4 (c+d x)^{7/2} a^5}{(a+b x)^{7/2}}-\frac {33 d^5 (c+d x)^{5/2} a^5}{(a+b x)^{5/2}}-\frac {135 b^3 c d^2 (c+d x)^{9/2} a^4}{(a+b x)^{9/2}}+\frac {360 b^2 c d^3 (c+d x)^{7/2} a^4}{(a+b x)^{7/2}}-\frac {305 b c d^4 (c+d x)^{5/2} a^4}{(a+b x)^{5/2}}+\frac {120 c d^5 (c+d x)^{3/2} a^4}{(a+b x)^{3/2}}+\frac {75 b^4 c^2 d (c+d x)^{9/2} a^3}{(a+b x)^{9/2}}-\frac {360 b^2 c^2 d^3 (c+d x)^{5/2} a^3}{(a+b x)^{5/2}}+\frac {280 b c^2 d^4 (c+d x)^{3/2} a^3}{(a+b x)^{3/2}}-\frac {75 c^2 d^5 \sqrt {c+d x} a^3}{\sqrt {a+b x}}+\frac {75 b^5 c^3 (c+d x)^{9/2} a^2}{(a+b x)^{9/2}}-\frac {280 b^4 c^3 d (c+d x)^{7/2} a^2}{(a+b x)^{7/2}}+\frac {360 b^3 c^3 d^2 (c+d x)^{5/2} a^2}{(a+b x)^{5/2}}-\frac {75 b c^3 d^4 \sqrt {c+d x} a^2}{\sqrt {a+b x}}-\frac {120 b^5 c^4 (c+d x)^{7/2} a}{(a+b x)^{7/2}}+\frac {305 b^4 c^4 d (c+d x)^{5/2} a}{(a+b x)^{5/2}}-\frac {360 b^3 c^4 d^2 (c+d x)^{3/2} a}{(a+b x)^{3/2}}+\frac {135 b^2 c^4 d^3 \sqrt {c+d x} a}{\sqrt {a+b x}}+\frac {33 b^5 c^5 (c+d x)^{5/2}}{(a+b x)^{5/2}}-\frac {40 b^4 c^5 d (c+d x)^{3/2}}{(a+b x)^{3/2}}+\frac {15 b^3 c^5 d^2 \sqrt {c+d x}}{\sqrt {a+b x}}}{24 \left (c-\frac {a (c+d x)}{a+b x}\right )^2 \left (\frac {b (c+d x)}{a+b x}-d\right )^3}-\frac {5}{4} \left (3 \sqrt {c} d^2 a^{5/2}+10 b c^{3/2} d a^{3/2}+3 b^2 c^{5/2} \sqrt {a}\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {5 \left (b^3 c^3+15 a b^2 d c^2+15 a^2 b d^2 c+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 \sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 23.19, size = 1469, normalized size = 4.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 20.40, size = 1336, normalized size = 4.19
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 850, normalized size = 2.66 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-90 \sqrt {b d}\, a^{3} c \,d^{2} x^{2} \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 )+15 \sqrt {a c}\, a^{3} d^{3} x^{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 )-300 \sqrt {b d}\, a^{2} b \,c^{2} d \,x^{2} \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 )+225 \sqrt {a c}\, a^{2} b c \,d^{2} x^{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 )-90 \sqrt {b d}\, a \,b^{2} c^{3} x^{2} \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 )+225 \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{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 )+15 \sqrt {a c}\, b^{3} c^{3} x^{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 )+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^{4}+52 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a b \,d^{2} x^{3}+52 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} c d \,x^{3}+66 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} d^{2} x^{2}+244 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a b c d \,x^{2}+66 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} c^{2} x^{2}-108 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} c d x -108 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a b \,c^{2} x -24 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} c^{2}\right )}{48 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, x^{2}} \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 )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^3} \,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 {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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