Optimal. Leaf size=329 \[ \frac{315 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{105 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{3 e}{8 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.181807, 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} \[ \frac{315 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}+\frac{105 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}-\frac{21 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}+\frac{3 e}{8 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 646
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^5 (d+e x)^{3/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (9 b^3 e \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^4 (d+e x)^{3/2}} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (21 b^2 e^2 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 (d+e x)^{3/2}} \, dx}{16 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (105 b e^3 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 (d+e x)^{3/2}} \, dx}{64 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3}{64 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3}{64 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 b e^4 \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{128 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3}{64 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (315 b e^3 \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 )}{64 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3}{64 (b d-a e)^4 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{4 (b d-a e) (a+b x)^3 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e}{8 (b d-a e)^2 (a+b x)^2 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2}{32 (b d-a e)^3 (a+b x) \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{315 e^4 (a+b x)}{64 (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{315 \sqrt{b} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0252979, size = 65, normalized size = 0.2 \[ \frac{2 e^4 (a+b x) \, _2F_1\left (-\frac{1}{2},5;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{\sqrt{(a+b x)^2} \sqrt{d+e x} (b d-a e)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.285, size = 602, normalized size = 1.8 \begin{align*} -{\frac{bx+a}{64\, \left ( ae-bd \right ) ^{5}} \left ( 315\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{4}{b}^{5}{e}^{4}+1260\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{3}a{b}^{4}{e}^{4}+315\,\sqrt{ \left ( ae-bd \right ) b}{x}^{4}{b}^{4}{e}^{4}+1890\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{x}^{2}{a}^{2}{b}^{3}{e}^{4}+1155\,\sqrt{ \left ( ae-bd \right ) b}{x}^{3}a{b}^{3}{e}^{4}+105\,\sqrt{ \left ( ae-bd \right ) b}{x}^{3}{b}^{4}d{e}^{3}+1260\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}x{a}^{3}{b}^{2}{e}^{4}+1533\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{a}^{2}{b}^{2}{e}^{4}+399\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}a{b}^{3}d{e}^{3}-42\,\sqrt{ \left ( ae-bd \right ) b}{x}^{2}{b}^{4}{d}^{2}{e}^{2}+315\,\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) \sqrt{ex+d}{a}^{4}b{e}^{4}+837\,\sqrt{ \left ( ae-bd \right ) b}x{a}^{3}b{e}^{4}+555\,\sqrt{ \left ( ae-bd \right ) b}x{a}^{2}{b}^{2}d{e}^{3}-156\,\sqrt{ \left ( ae-bd \right ) b}xa{b}^{3}{d}^{2}{e}^{2}+24\,\sqrt{ \left ( ae-bd \right ) b}x{b}^{4}{d}^{3}e+128\,\sqrt{ \left ( ae-bd \right ) b}{a}^{4}{e}^{4}+325\,\sqrt{ \left ( ae-bd \right ) b}{a}^{3}bd{e}^{3}-210\,\sqrt{ \left ( ae-bd \right ) b}{a}^{2}{b}^{2}{d}^{2}{e}^{2}+88\,\sqrt{ \left ( ae-bd \right ) b}a{b}^{3}{d}^{3}e-16\,\sqrt{ \left ( ae-bd \right ) b}{b}^{4}{d}^{4} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.01033, size = 3549, normalized size = 10.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34138, size = 1129, normalized size = 3.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]