Optimal. Leaf size=197 \[ -\frac {a^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{4 d} \]
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Rubi [A] time = 0.20, antiderivative size = 197, 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 = {97, 154, 157, 63, 217, 206, 93, 208} \[ -\frac {\sqrt {b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}}-\frac {a^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{4 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 97
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx &=-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\int \frac {(a+b x)^{3/2} \left (\frac {1}{2} (5 b c+a d)+3 b d x\right )}{x \sqrt {c+d x}} \, dx\\ &=\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {\int \frac {\sqrt {a+b x} \left (a d (5 b c+a d)+\frac {1}{2} b d (b c+11 a d) x\right )}{x \sqrt {c+d x}} \, dx}{2 d}\\ &=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {\int \frac {a^2 d^2 (5 b c+a d)-\frac {1}{4} b d \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^2}\\ &=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\frac {1}{2} \left (a^2 (5 b c+a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (b \left (b^2 c^2-10 a b c d-15 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d}\\ &=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}+\left (a^2 (5 b c+a d)\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^2 c^2-10 a b c d-15 a^2 d^2\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 )}{4 d}\\ &=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}-\frac {a^{3/2} (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\left (b^2 c^2-10 a b c d-15 a^2 d^2\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 )}{4 d}\\ &=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}-\frac {a^{3/2} (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 204, normalized size = 1.04 \[ -\frac {a^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b c-a d} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 d^{3/2} \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d+9 a b d x+b^2 x (c+2 d x)\right )}{4 d x} \]
Antiderivative was successfully verified.
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fricas [A] time = 6.03, size = 1074, normalized size = 5.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.47, size = 578, normalized size = 2.93 \[ \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left | b \right |} + \frac {b c d {\left | b \right |} + 7 \, a d^{2} {\left | b \right |}}{d^{2}}\right )} - \frac {8 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c {\left | b \right |} + \sqrt {b d} a^{3} b d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {16 \, {\left (\sqrt {b d} a^{2} b^{4} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} c {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}} + \frac {{\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 10 \, \sqrt {b d} a b c d {\left | b \right |} - 15 \, \sqrt {b d} a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2}}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 504, normalized size = 2.56 \[ \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-4 \sqrt {b d}\, a^{3} d^{2} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-20 \sqrt {b d}\, a^{2} b c d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+15 \sqrt {a c}\, a^{2} b \,d^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+10 \sqrt {a c}\, a \,b^{2} c d x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-\sqrt {a c}\, b^{3} c^{2} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d \,x^{2}+18 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a b d x +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} c x -8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} d \right )}{8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, d x} \]
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 = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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