Optimal. Leaf size=329 \[ \frac{63 b^2 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2 (a+b x)}{20 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{9 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2} \]
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Rubi [A] time = 0.160639, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {646, 51, 63, 208} \[ \frac{63 b^2 e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{21 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{63 e^2 (a+b x)}{20 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac{9 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 646
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (9 b e \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 (d+e x)^{7/2}} \, dx}{4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{8 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 b e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{8 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 b^3 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{8 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (63 b^3 e \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{9 e}{4 (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 e^2 (a+b x)}{20 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 b e^2 (a+b x)}{4 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{63 b^2 e^2 (a+b x)}{4 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{63 b^{5/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0273192, size = 67, normalized size = 0.2 \[ \frac{2 e^2 (a+b x) \, _2F_1\left (-\frac{5}{2},3;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{5 \sqrt{(a+b x)^2} (d+e x)^{5/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.307, size = 518, normalized size = 1.6 \begin{align*} -{\frac{bx+a}{20\, \left ( ae-bd \right ) ^{5}} \left ( 315\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{5/2}{x}^{2}{b}^{5}{e}^{2}+630\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{5/2}xa{b}^{4}{e}^{2}+315\,\sqrt{ \left ( ae-bd \right ) b}{x}^{4}{b}^{4}{e}^{4}+315\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \left ( ex+d \right ) ^{5/2}{a}^{2}{b}^{3}{e}^{2}+525\,\sqrt{ \left ( ae-bd \right ) b}{x}^{3}a{b}^{3}{e}^{4}+735\,\sqrt{ \left ( ae-bd \right ) b}{x}^{3}{b}^{4}d{e}^{3}+168\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{a}^{2}{b}^{2}{e}^{4}+1239\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}a{b}^{3}d{e}^{3}+483\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{4}{d}^{2}{e}^{2}-24\,\sqrt{ \left ( ae-bd \right ) b}x{a}^{3}b{e}^{4}+408\,\sqrt{ \left ( ae-bd \right ) b}x{a}^{2}{b}^{2}d{e}^{3}+831\,\sqrt{ \left ( ae-bd \right ) b}xa{b}^{3}{d}^{2}{e}^{2}+45\,\sqrt{ \left ( ae-bd \right ) b}x{b}^{4}{d}^{3}e+8\,\sqrt{ \left ( ae-bd \right ) b}{a}^{4}{e}^{4}-56\,\sqrt{ \left ( ae-bd \right ) b}{a}^{3}bd{e}^{3}+288\,\sqrt{ \left ( ae-bd \right ) b}{a}^{2}{b}^{2}{d}^{2}{e}^{2}+85\,\sqrt{ \left ( ae-bd \right ) b}a{b}^{3}{d}^{3}e-10\,\sqrt{ \left ( ae-bd \right ) b}{b}^{4}{d}^{4} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91255, size = 3753, normalized size = 11.41 \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(-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 [B] time = 1.29158, size = 1046, normalized size = 3.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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