Optimal. Leaf size=147 \[ -\frac{5 d^2 \sqrt{c+d x}}{8 (a+b x) (b c-a d)^3}+\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}+\frac{5 d \sqrt{c+d x}}{12 (a+b x)^2 (b c-a d)^2}-\frac{\sqrt{c+d x}}{3 (a+b x)^3 (b c-a d)} \]
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Rubi [A] time = 0.0504401, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {51, 63, 208} \[ -\frac{5 d^2 \sqrt{c+d x}}{8 (a+b x) (b c-a d)^3}+\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}+\frac{5 d \sqrt{c+d x}}{12 (a+b x)^2 (b c-a d)^2}-\frac{\sqrt{c+d x}}{3 (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^4 \sqrt{c+d x}} \, dx &=-\frac{\sqrt{c+d x}}{3 (b c-a d) (a+b x)^3}-\frac{(5 d) \int \frac{1}{(a+b x)^3 \sqrt{c+d x}} \, dx}{6 (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{3 (b c-a d) (a+b x)^3}+\frac{5 d \sqrt{c+d x}}{12 (b c-a d)^2 (a+b x)^2}+\frac{\left (5 d^2\right ) \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{8 (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{3 (b c-a d) (a+b x)^3}+\frac{5 d \sqrt{c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac{5 d^2 \sqrt{c+d x}}{8 (b c-a d)^3 (a+b x)}-\frac{\left (5 d^3\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{16 (b c-a d)^3}\\ &=-\frac{\sqrt{c+d x}}{3 (b c-a d) (a+b x)^3}+\frac{5 d \sqrt{c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac{5 d^2 \sqrt{c+d x}}{8 (b c-a d)^3 (a+b x)}-\frac{\left (5 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{8 (b c-a d)^3}\\ &=-\frac{\sqrt{c+d x}}{3 (b c-a d) (a+b x)^3}+\frac{5 d \sqrt{c+d x}}{12 (b c-a d)^2 (a+b x)^2}-\frac{5 d^2 \sqrt{c+d x}}{8 (b c-a d)^3 (a+b x)}+\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0115655, size = 50, normalized size = 0.34 \[ \frac{2 d^3 \sqrt{c+d x} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};-\frac{b (c+d x)}{a d-b c}\right )}{(a d-b c)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 147, normalized size = 1. \begin{align*}{\frac{{d}^{3}}{ \left ( 3\,ad-3\,bc \right ) \left ( bdx+ad \right ) ^{3}}\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{12\, \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{8\, \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26014, size = 1805, normalized size = 12.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07195, size = 312, normalized size = 2.12 \begin{align*} -\frac{5 \, d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{15 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} + 33 \, \sqrt{d x + c} b^{2} c^{2} d^{3} + 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} - 66 \, \sqrt{d x + c} a b c d^{4} + 33 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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