Optimal. Leaf size=154 \[ -\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}+\frac {15 \sqrt {c+d x} (b c-a d)^2}{4 a^3 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (b c-a d)}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}} \]
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Rubi [A] time = 0.07, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac {5 (c+d x)^{3/2} (b c-a d)}{4 a^2 x \sqrt {a+b x}}+\frac {15 \sqrt {c+d x} (b c-a d)^2}{4 a^3 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx &=-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {(5 (b c-a d)) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{4 a}\\ &=\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}+\frac {\left (15 c (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^3}\\ &=\frac {15 (b c-a d)^2 \sqrt {c+d x}}{4 a^3 \sqrt {a+b x}}+\frac {5 (b c-a d) (c+d x)^{3/2}}{4 a^2 x \sqrt {a+b x}}-\frac {(c+d x)^{5/2}}{2 a x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 130, normalized size = 0.84 \[ \frac {\sqrt {c+d x} \left (a^2 \left (-2 c^2-9 c d x+8 d^2 x^2\right )+5 a b c x (c-5 d x)+15 b^2 c^2 x^2\right )}{4 a^3 x^2 \sqrt {a+b x}}-\frac {15 \sqrt {c} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.64, size = 481, normalized size = 3.12 \[ \left [\frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (15 \, b^{2} c^{2} - 25 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{2} - {\left (5 \, a b c^{2} - 9 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 21.61, size = 1206, normalized size = 7.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 507, normalized size = 3.29 \[ -\frac {\sqrt {d x +c}\, \left (15 a^{2} b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-30 a \,b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 b^{3} c^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 a^{3} c \,d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-30 a^{2} b \,c^{2} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+15 a \,b^{2} c^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}+50 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}+18 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} c d x -10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a b \,c^{2} x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} c^{2}\right )}{8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \sqrt {b x +a}\, a^{3} x^{2}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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