Optimal. Leaf size=226 \[ \frac {5 (b c-7 a d) (b c-a d)^2 \sqrt {a+b x}}{8 a c^4 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 226, 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-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}}+\frac {5 \sqrt {a+b x} (b c-7 a d) (b c-a d)^2}{8 a c^4 \sqrt {c+d x}}-\frac {5 (a+b x)^{3/2} (b c-7 a d) (b c-a d)}{24 a c^3 x \sqrt {c+d x}}-\frac {(a+b x)^{5/2} (b c-7 a d)}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 96
Rule 98
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx &=-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}-\frac {\left (-\frac {b c}{2}+\frac {7 a d}{2}\right ) \int \frac {(a+b x)^{5/2}}{x^3 (c+d x)^{3/2}} \, dx}{3 a c}\\ &=-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}+\frac {(5 (b c-7 a d) (b c-a d)) \int \frac {(a+b x)^{3/2}}{x^2 (c+d x)^{3/2}} \, dx}{24 a c^2}\\ &=-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}+\frac {\left (5 (b c-7 a d) (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}} \, dx}{16 a c^3}\\ &=\frac {5 (b c-7 a d) (b c-a d)^2 \sqrt {a+b x}}{8 a c^4 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}+\frac {\left (5 (b c-7 a d) (b c-a d)^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 c^4}\\ &=\frac {5 (b c-7 a d) (b c-a d)^2 \sqrt {a+b x}}{8 a c^4 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}+\frac {\left (5 (b c-7 a d) (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 c^4}\\ &=\frac {5 (b c-7 a d) (b c-a d)^2 \sqrt {a+b x}}{8 a c^4 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.37, size = 171, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {a+b x} \left (3 b^2 c^2 x^2 (11 c+27 d x)+2 a b c x \left (13 c^2-34 c d x-95 d^2 x^2\right )+a^2 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )\right )}{24 c^4 x^3 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 (-b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(703\) vs.
\(2(188)=376\).
time = 0.07, size = 704, normalized size = 3.12
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (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^{3} d^{4} 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^{2} b c \,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 \,b^{2} c^{2} d^{2} 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 ) b^{3} c^{3} 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 ) a^{3} c \,d^{3} 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 \,c^{2} d^{2} 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 \,b^{2} c^{3} d \,x^{3}-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 ) b^{3} c^{4} x^{3}-210 a^{2} d^{3} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+380 a b c \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-162 b^{2} c^{2} d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-70 a^{2} c \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+136 a b \,c^{2} d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-66 b^{2} c^{3} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+28 a^{2} c^{2} d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-52 a b \,c^{3} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-16 a^{2} c^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{48 c^{4} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}\, \sqrt {d x +c}}\) | \(704\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.00, size = 632, normalized size = 2.80 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c 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^{3} c^{4} + {\left (81 \, a b^{2} c^{3} d - 190 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} + {\left (33 \, a b^{2} c^{4} - 68 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{4} - 7 \, a^{3} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a c^{5} d x^{4} + a c^{6} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c 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^{3} c^{4} + {\left (81 \, a b^{2} c^{3} d - 190 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} + {\left (33 \, a b^{2} c^{4} - 68 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{4} - 7 \, a^{3} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{5} d x^{4} + a c^{6} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2200 vs.
\(2 (188) = 376\).
time = 10.16, size = 2200, normalized size = 9.73 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^4\,{\left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________