Optimal. Leaf size=438 \[ \frac{a^2 b C d^2+a^3 \left (-d^2\right ) D+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left (-\left (7 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right )\right )}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b d (C d-4 c D)+a^3 d^2 D+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{c+d x} \left (a^2 b (12 c D+C d)-5 a^3 d D-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{4 b (a+b x) (b c-a d)^4} \]
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Rubi [A] time = 1.21388, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1621, 897, 1259, 1261, 208} \[ \frac{a^2 b C d^2+a^3 \left (-d^2\right ) D+a b^2 \left (-3 B d^2-6 c^2 D+4 c C d\right )+b^3 \left (7 A d^2-4 B c d+2 c^2 C\right )}{b^2 \sqrt{c+d x} (b c-a d)^4}-\frac{-3 a^2 b C d^3+3 a^3 d^3 D+3 a b^2 B d^3+b^3 \left (-\left (7 A d^3-4 B c d^2+4 c^2 C d-4 c^3 D\right )\right )}{6 b^3 d (c+d x)^{3/2} (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b d (C d-4 c D)+a^3 d^2 D+3 a b^2 \left (-5 B d^2-8 c^2 D+8 c C d\right )+b^3 \left (35 A d^2-20 B c d+8 c^2 C\right )\right )}{4 b^{3/2} (b c-a d)^{9/2}}-\frac{A b^3-a \left (a^2 D-a b C+b^2 B\right )}{2 b^3 (a+b x)^2 (c+d x)^{3/2} (b c-a d)}-\frac{\sqrt{c+d x} \left (a^2 b (12 c D+C d)-5 a^3 d D-a b^2 (8 c C-3 B d)+b^3 (4 B c-7 A d)\right )}{4 b (a+b x) (b c-a d)^4} \]
Antiderivative was successfully verified.
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Rule 1621
Rule 897
Rule 1259
Rule 1261
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2+D x^3}{(a+b x)^3 (c+d x)^{5/2}} \, dx &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac{\int \frac{-\frac{b^3 (4 B c-7 A d)-a b^2 (4 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-4 c D)}{2 b^3}-\frac{2 (b c-a d) (b C-a D) x}{b^2}-2 \left (c-\frac{a d}{b}\right ) D x^2}{(a+b x)^2 (c+d x)^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{-2 c^2 \left (c-\frac{a d}{b}\right ) D+\frac{2 c d (b c-a d) (b C-a D)}{b^2}-\frac{d^2 \left (b^3 (4 B c-7 A d)-a b^2 (4 c C-3 B d)+3 a^3 d D-a^2 b (3 C d-4 c D)\right )}{2 b^3}}{d^2}-\frac{\left (-4 c \left (c-\frac{a d}{b}\right ) D+\frac{2 d (b c-a d) (b C-a D)}{b^2}\right ) x^2}{d^2}-\frac{2 \left (c-\frac{a d}{b}\right ) D x^4}{d^2}}{x^4 \left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )^2} \, dx,x,\sqrt{c+d x}\right )}{d (b c-a d)}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac{\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt{c+d x}}{4 b (b c-a d)^4 (a+b x)}+\frac{d^4 \operatorname{Subst}\left (\int \frac{-\frac{(b c-a d)^2 \left (3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )\right )}{b d^6}-\frac{(b c-a d) \left (a^2 b C d^3-a^3 d^3 D-a b^2 d \left (8 c C d-3 B d^2-12 c^2 D\right )+b^3 \left (4 B c d^2-7 A d^3-4 c^3 D\right )\right ) x^2}{d^6}-\frac{b \left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) x^4}{2 d^4}}{x^4 \left (\frac{-b c+a d}{d}+\frac{b x^2}{d}\right )} \, dx,x,\sqrt{c+d x}\right )}{2 b^2 (b c-a d)^4}\\ &=-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}-\frac{\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt{c+d x}}{4 b (b c-a d)^4 (a+b x)}+\frac{d^4 \operatorname{Subst}\left (\int \left (\frac{(b c-a d) \left (3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )\right )}{b d^5 x^4}+\frac{2 \left (-a^2 b C d^2-b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )+a^3 d^2 D-a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )\right )}{d^4 x^2}+\frac{b \left (-b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )-a^3 d^2 D-3 a^2 b d (C d-4 c D)-3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right )}{2 d^4 \left (b c-a d-b x^2\right )}\right ) \, dx,x,\sqrt{c+d x}\right )}{2 b^2 (b c-a d)^4}\\ &=-\frac{3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac{a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt{c+d x}}-\frac{\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt{c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac{\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b c-a d-b x^2} \, dx,x,\sqrt{c+d x}\right )}{4 b (b c-a d)^4}\\ &=-\frac{3 a b^2 B d^3-3 a^2 b C d^3+3 a^3 d^3 D-b^3 \left (4 c^2 C d-4 B c d^2+7 A d^3-4 c^3 D\right )}{6 b^3 d (b c-a d)^3 (c+d x)^{3/2}}-\frac{A b^3-a \left (b^2 B-a b C+a^2 D\right )}{2 b^3 (b c-a d) (a+b x)^2 (c+d x)^{3/2}}+\frac{a^2 b C d^2+b^3 \left (2 c^2 C-4 B c d+7 A d^2\right )-a^3 d^2 D+a b^2 \left (4 c C d-3 B d^2-6 c^2 D\right )}{b^2 (b c-a d)^4 \sqrt{c+d x}}-\frac{\left (b^3 (4 B c-7 A d)-a b^2 (8 c C-3 B d)-5 a^3 d D+a^2 b (C d+12 c D)\right ) \sqrt{c+d x}}{4 b (b c-a d)^4 (a+b x)}-\frac{\left (b^3 \left (8 c^2 C-20 B c d+35 A d^2\right )+a^3 d^2 D+3 a^2 b d (C d-4 c D)+3 a b^2 \left (8 c C d-5 B d^2-8 c^2 D\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{9/2}}\\ \end{align*}
Mathematica [A] time = 2.15449, size = 536, normalized size = 1.22 \[ -\frac{\sqrt{c+d x} \left (3 a^2 b c D+a^3 (-d) D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right )}{b (a+b x) (b c-a d)^4}+\frac{\sqrt{c+d x} \left (a \left (a^2 D-a b C+b^2 B\right )-A b^3\right )}{2 b (a+b x)^2 (b c-a d)^3}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (3 a^2 b c D+a^3 (-d) D+a b^2 (B d-2 c C)+b^3 (B c-2 A d)\right )}{b^{3/2} (b c-a d)^{9/2}}-\frac{3 d \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \left (d (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )-\sqrt{b} \sqrt{c+d x} \sqrt{b c-a d}\right )}{4 b^{3/2} (a+b x) (b c-a d)^{9/2}}+\frac{2 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{3 d (c+d x)^{3/2} (b c-a d)^3}+\frac{2 \left (b \left (3 A d^2-2 B c d+c^2 C\right )-a \left (B d^2+3 c^2 D-2 c C d\right )\right )}{\sqrt{c+d x} (b c-a d)^4}-\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right ) \left (b \left (3 A d^2-2 B c d+c^2 C\right )-a \left (B d^2+3 c^2 D-2 c C d\right )\right )}{(b c-a d)^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 1376, normalized size = 3.1 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [A] time = 2.29661, size = 1035, normalized size = 2.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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