Optimal. Leaf size=303 \[ -\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.330066, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {742, 832, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 742
Rule 832
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{2 \int \frac{\sqrt{d+e x} \left (\frac{1}{2} d (5 c d-b e)+2 e (2 c d-b e) x\right )}{\sqrt{b x+c x^2}} \, dx}{5 c}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{4 \int \frac{\frac{1}{4} d \left (15 c^2 d^2-11 b c d e+4 b^2 e^2\right )+\frac{1}{4} e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}-\frac{(4 d (c d-b e) (2 c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{15 c^2}+\frac{\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}-\frac{\left (4 d (c d-b e) (2 c d-b e) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{15 c^2 \sqrt{b x+c x^2}}+\frac{\left (\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{15 c^2 \sqrt{b x+c x^2}}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{\left (\left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{x} \sqrt{1+\frac{c x}{b}}} \, dx}{15 c^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (4 d (c d-b e) (2 c d-b e) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}} \, dx}{15 c^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{8 e (2 c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 e (d+e x)^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{2 \sqrt{-b} \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{8 \sqrt{-b} d (c d-b e) (2 c d-b e) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.28467, size = 314, normalized size = 1.04 \[ \frac{2 \sqrt{x} \left (-\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-27 b^2 c d e^2+8 b^3 e^3+34 b c^2 d^2 e-15 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )}{b}+\frac{(b+c x) (d+e x) \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+e \sqrt{x} (b+c x) (d+e x) (-4 b e+11 c d+3 c e x)\right )}{15 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.272, size = 682, normalized size = 2.3 \begin{align*} -{\frac{2}{15\,x \left ( ce{x}^{2}+bxe+cdx+bd \right ){c}^{4}}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ( 4\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}-12\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e+8\,\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}+8\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{4}{e}^{3}-31\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{3}cd{e}^{2}+46\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}{b}^{2}{c}^{2}{d}^{2}e-23\,\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) \sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}b{c}^{3}{d}^{3}-3\,{x}^{4}{c}^{4}{e}^{3}+{x}^{3}b{c}^{3}{e}^{3}-14\,{x}^{3}{c}^{4}d{e}^{2}+4\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-10\,{x}^{2}b{c}^{3}d{e}^{2}-11\,{x}^{2}{c}^{4}{d}^{2}e+4\,x{b}^{2}{c}^{2}d{e}^{2}-11\,xb{c}^{3}{d}^{2}e \right ) } \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{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \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{\left (d + e x\right )^{\frac{5}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]