Optimal. Leaf size=230 \[ -\frac {5 (b c-a d)^2 (7 b c-a d) \sqrt {c+d x}}{8 a^4 c \sqrt {a+b x}}-\frac {5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt {a+b x}}+\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}+\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} \sqrt {c}} \]
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Rubi [A]
time = 0.08, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {98, 96, 95, 214}
\begin {gather*} \frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} \sqrt {c}}-\frac {5 \sqrt {c+d x} (b c-a d)^2 (7 b c-a d)}{8 a^4 c \sqrt {a+b x}}-\frac {5 (c+d x)^{3/2} (b c-a d) (7 b c-a d)}{24 a^3 c x \sqrt {a+b x}}+\frac {(c+d x)^{5/2} (7 b c-a d)}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^4 (a+b x)^{3/2}} \, dx &=-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}-\frac {\left (\frac {7 b c}{2}-\frac {a d}{2}\right ) \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx}{3 a c}\\ &=\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}+\frac {(5 (b c-a d) (7 b c-a d)) \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{24 a^2 c}\\ &=-\frac {5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt {a+b x}}+\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{16 a^3 c}\\ &=-\frac {5 (b c-a d)^2 (7 b c-a d) \sqrt {c+d x}}{8 a^4 c \sqrt {a+b x}}-\frac {5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt {a+b x}}+\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^4}\\ &=-\frac {5 (b c-a d)^2 (7 b c-a d) \sqrt {c+d x}}{8 a^4 c \sqrt {a+b x}}-\frac {5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt {a+b x}}+\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}-\frac {\left (5 (b c-a d)^2 (7 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^4}\\ &=-\frac {5 (b c-a d)^2 (7 b c-a d) \sqrt {c+d x}}{8 a^4 c \sqrt {a+b x}}-\frac {5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt {a+b x}}+\frac {(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt {a+b x}}-\frac {(c+d x)^{7/2}}{3 a c x^3 \sqrt {a+b x}}+\frac {5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 169, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {c+d x} \left (105 b^3 c^2 x^3+5 a b^2 c x^2 (7 c-38 d x)+a^3 \left (8 c^2+26 c d x+33 d^2 x^2\right )+a^2 b x \left (-14 c^2-68 c d x+81 d^2 x^2\right )\right )}{24 a^4 x^3 \sqrt {a+b x}}-\frac {5 (b c-a d)^2 (-7 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 a^{9/2} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(703\) vs.
\(2(192)=384\).
time = 0.07, size = 704, normalized size = 3.06
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b \,d^{3} x^{4}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c \,d^{2} x^{4}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{2} d \,x^{4}-105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{3} x^{4}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{3} x^{3}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{2} x^{3}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d \,x^{3}-105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} x^{3}+162 a^{2} b \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-380 a \,b^{2} c d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+210 b^{3} c^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+66 a^{3} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-136 a^{2} b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+70 a \,b^{2} c^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+52 a^{3} c d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-28 a^{2} b \,c^{2} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+16 a^{3} c^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{48 a^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}\, \sqrt {b x +a}}\) | \(704\) |
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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Fricas [A]
time = 2.76, size = 636, normalized size = 2.77 \begin {gather*} \left [-\frac {15 \, {\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + {\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \sqrt {a c} \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 c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 190 \, a^{2} b^{2} c^{2} d + 81 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 68 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (7 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}, -\frac {15 \, {\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + {\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 190 \, a^{2} b^{2} c^{2} d + 81 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 68 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \, {\left (7 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2324 vs.
\(2 (192) = 384\).
time = 12.11, size = 2324, normalized size = 10.10 \begin {gather*} \text {Too large to display} \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 (c+d\,x\right )}^{5/2}}{x^4\,{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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