Optimal. Leaf size=241 \[ -\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}-\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]
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Rubi [A] time = 0.232663, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {98, 151, 152, 12, 93, 208} \[ -\frac{b \sqrt{c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt{a+b x}}+\frac{(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}+\frac{7 \sqrt{c+d x} (b c-a d)}{12 a^2 x^2 \sqrt{a+b x}}-\frac{\sqrt{c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 152
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx &=-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}-\frac{\int \frac{\frac{7}{2} c (b c-a d)+3 d (b c-a d) x}{x^3 (a+b x)^{3/2} \sqrt{c+d x}} \, dx}{3 a}\\ &=-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}+\frac{\int \frac{\frac{1}{4} c (35 b c-3 a d) (b c-a d)+7 b c d (b c-a d) x}{x^2 (a+b x)^{3/2} \sqrt{c+d x}} \, dx}{6 a^2 c}\\ &=-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}-\frac{(35 b c-3 a d) (b c-a d) \sqrt{c+d x}}{24 a^3 c x \sqrt{a+b x}}-\frac{\int \frac{\frac{3}{8} c (b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )+\frac{1}{4} b c d (35 b c-3 a d) (b c-a d) x}{x (a+b x)^{3/2} \sqrt{c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac{b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt{c+d x}}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}-\frac{(35 b c-3 a d) (b c-a d) \sqrt{c+d x}}{24 a^3 c x \sqrt{a+b x}}-\frac{\int \frac{3 c (b c-a d)^2 \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a^4 c^2 (b c-a d)}\\ &=-\frac{b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt{c+d x}}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}-\frac{(35 b c-3 a d) (b c-a d) \sqrt{c+d x}}{24 a^3 c x \sqrt{a+b x}}-\frac{\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a^4 c}\\ &=-\frac{b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt{c+d x}}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}-\frac{(35 b c-3 a d) (b c-a d) \sqrt{c+d x}}{24 a^3 c x \sqrt{a+b x}}-\frac{\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 a^4 c}\\ &=-\frac{b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt{c+d x}}{24 a^4 c \sqrt{a+b x}}-\frac{c \sqrt{c+d x}}{3 a x^3 \sqrt{a+b x}}+\frac{7 (b c-a d) \sqrt{c+d x}}{12 a^2 x^2 \sqrt{a+b x}}-\frac{(35 b c-3 a d) (b c-a d) \sqrt{c+d x}}{24 a^3 c x \sqrt{a+b x}}+\frac{(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.147491, size = 190, normalized size = 0.79 \[ \frac{\left (9 a^2 b c d^2+a^3 d^3-45 a b^2 c^2 d+35 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac{\sqrt{c+d x} \left (a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )+5 a b^2 c x^2 (7 c-20 d x)+105 b^3 c^2 x^3\right )}{24 a^4 c x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 707, normalized size = 2.9 \begin{align*} \text{result too large to display} \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 [A] time = 19.5063, size = 1374, normalized size = 5.7 \begin{align*} \left [\frac{3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, 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} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} +{\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}, -\frac{3 \,{\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} +{\left (35 \, a b^{3} c^{3} - 45 \, 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} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} +{\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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