Optimal. Leaf size=163 \[ \frac {c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}-\frac {\sqrt {c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt {a+b x}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {98, 150, 157, 63, 217, 206, 93, 208} \[ \frac {c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}-\frac {\sqrt {c+d x} (3 b c-2 a d) (b c-a d)}{a^2 b \sqrt {a+b x}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 98
Rule 150
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{3/2}} \, dx &=-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}-\frac {\int \frac {\sqrt {c+d x} \left (\frac {1}{2} c (3 b c-5 a d)-a d^2 x\right )}{x (a+b x)^{3/2}} \, dx}{a}\\ &=-\frac {(3 b c-2 a d) (b c-a d) \sqrt {c+d x}}{a^2 b \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {2 \int \frac {-\frac {1}{4} b c^2 (3 b c-5 a d)+\frac {1}{2} a^2 d^3 x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{a^2 b}\\ &=-\frac {(3 b c-2 a d) (b c-a d) \sqrt {c+d x}}{a^2 b \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b}-\frac {\left (c^2 (3 b c-5 a d)\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 a^2}\\ &=-\frac {(3 b c-2 a d) (b c-a d) \sqrt {c+d x}}{a^2 b \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {\left (2 d^3\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 )}{b^2}-\frac {\left (c^2 (3 b c-5 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 )}{a^2}\\ &=-\frac {(3 b c-2 a d) (b c-a d) \sqrt {c+d x}}{a^2 b \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {\left (2 d^3\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 )}{b^2}\\ &=-\frac {(3 b c-2 a d) (b c-a d) \sqrt {c+d x}}{a^2 b \sqrt {a+b x}}-\frac {c (c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {c^{3/2} (3 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 3.42, size = 934, normalized size = 5.73 \[ -\frac {\sqrt {c+d x} \left (-3 d^{7/2} x \sqrt {a+b x} (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) a^{7/2}-2 d^3 \sqrt {b c-a d} x (c+d x)^2 \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right ) a^{7/2}+3 d^3 \sqrt {b c-a d} x (c+d x)^2 \sqrt {\frac {b (c+d x)}{b c-a d}} a^{7/2}+\frac {d^3 (b c-a d)^{5/2} x^2 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} a^{5/2}}{b}-\frac {7 c d^2 (b c-a d)^{5/2} x \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} a^{5/2}}{b}+8 b c d^{5/2} x \sqrt {a+b x} (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) a^{5/2}+8 b c d^2 \sqrt {b c-a d} x (c+d x)^2 \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right ) a^{5/2}+2 c^3 (b c-a d)^{5/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} a^{3/2}-3 c d^2 (b c-a d)^{5/2} x^2 \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} a^{3/2}-2 c^2 d (b c-a d)^{5/2} x \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} a^{3/2}-9 b^2 c^2 d^{3/2} x \sqrt {a+b x} (c+d x)^2 \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) a^{3/2}-6 b^2 c^2 d \sqrt {b c-a d} x (c+d x)^2 \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {d (a+b x)}{a d-b c}\right ) a^{3/2}+10 b c^{5/2} d (b c-a d)^{3/2} x \sqrt {a+b x} \sqrt {c+d x} \left (\frac {b (c+d x)}{b c-a d}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) a+6 b c^3 (b c-a d)^{5/2} x \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2} \sqrt {a}-6 b^3 c^{7/2} \sqrt {b c-a d} x \sqrt {a+b x} (c+d x)^{3/2} \sqrt {\frac {b (c+d x)}{b c-a d}} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )}{2 a^{5/2} c (b c-a d)^{5/2} x \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 3.89, size = 1234, normalized size = 7.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 502, normalized size = 3.08 \[ \frac {\sqrt {d x +c}\, \left (2 \sqrt {a c}\, a^{2} b \,d^{3} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-5 \sqrt {b d}\, a \,b^{2} c^{2} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 \sqrt {b d}\, b^{3} c^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+2 \sqrt {a c}\, a^{3} d^{3} x \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-5 \sqrt {b d}\, a^{2} b \,c^{2} d x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+3 \sqrt {b d}\, a \,b^{2} c^{3} x \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-4 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x +8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x -2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a b \,c^{2}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {b x +a}\, a^{2} b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^2\,{\left (a+b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________