Optimal. Leaf size=100 \[ -\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}-\frac {3 d \sqrt {c+d x}}{4 b^2 (a+b x)}-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 63, 208} \begin {gather*} -\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}-\frac {3 d \sqrt {c+d x}}{4 b^2 (a+b x)}-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^3} \, dx &=-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac {3 d \sqrt {c+d x}}{4 b^2 (a+b x)}-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2}+\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 b^2}\\ &=-\frac {3 d \sqrt {c+d x}}{4 b^2 (a+b x)}-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2}+\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 b^2}\\ &=-\frac {3 d \sqrt {c+d x}}{4 b^2 (a+b x)}-\frac {(c+d x)^{3/2}}{2 b (a+b x)^2}-\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{5/2} \sqrt {b c-a d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 90, normalized size = 0.90 \begin {gather*} \frac {3 d^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {a d-b c}}\right )}{4 b^{5/2} \sqrt {a d-b c}}-\frac {\sqrt {c+d x} (3 a d+2 b c+5 b d x)}{4 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.38, size = 116, normalized size = 1.16 \begin {gather*} -\frac {3 d^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{4 b^{5/2} \sqrt {a d-b c}}-\frac {d^2 \sqrt {c+d x} (3 a d+5 b (c+d x)-3 b c)}{4 b^2 (a d+b (c+d x)-b c)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.00, size = 383, normalized size = 3.83 \begin {gather*} \left [\frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + {\left (b^{6} c - a b^{5} d\right )} x^{2} + 2 \, {\left (a b^{5} c - a^{2} b^{4} d\right )} x\right )}}, \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} + 5 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c - a^{3} b^{3} d + {\left (b^{6} c - a b^{5} d\right )} x^{2} + 2 \, {\left (a b^{5} c - a^{2} b^{4} d\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.36, size = 108, normalized size = 1.08 \begin {gather*} \frac {3 \, d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} b^{2}} - \frac {5 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} - 3 \, \sqrt {d x + c} b c d^{2} + 3 \, \sqrt {d x + c} a d^{3}}{4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 121, normalized size = 1.21 \begin {gather*} -\frac {3 \sqrt {d x +c}\, a \,d^{3}}{4 \left (b d x +a d \right )^{2} b^{2}}+\frac {3 \sqrt {d x +c}\, c \,d^{2}}{4 \left (b d x +a d \right )^{2} b}-\frac {5 \left (d x +c \right )^{\frac {3}{2}} d^{2}}{4 \left (b d x +a d \right )^{2} b}+\frac {3 d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \sqrt {\left (a d -b c \right ) b}\, b^{2}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 0.28, size = 135, normalized size = 1.35 \begin {gather*} \frac {3\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{5/2}\,\sqrt {a\,d-b\,c}}-\frac {\frac {5\,d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,b}+\frac {3\,d^2\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{4\,b^2}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] 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]
________________________________________________________________________________________