Optimal. Leaf size=376 \[ \frac{8 a^{5/4} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a c \sqrt{a+c x^2} (63 A-25 B x)}{105 e^3 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (25 a B-21 A c x)}{105 e^2 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A-5 B x)}{35 e (e x)^{5/2}}+\frac{48 a A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.962693, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{8 a^{5/4} c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{48 a^{5/4} A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 a c \sqrt{a+c x^2} (63 A-25 B x)}{105 e^3 \sqrt{e x}}-\frac{4 \left (a+c x^2\right )^{3/2} (25 a B-21 A c x)}{105 e^2 (e x)^{3/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A-5 B x)}{35 e (e x)^{5/2}}+\frac{48 a A c^{3/2} x \sqrt{a+c x^2}}{5 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(7/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 123.844, size = 369, normalized size = 0.98 \[ - \frac{48 A a^{\frac{5}{4}} c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{48 A a c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 e^{3} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 a^{\frac{5}{4}} c^{\frac{3}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (63 A \sqrt{c} + 25 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{16 a c \left (\frac{63 A}{2} - \frac{25 B x}{2}\right ) \sqrt{a + c x^{2}}}{105 e^{3} \sqrt{e x}} - \frac{4 \left (\frac{7 A}{2} - \frac{5 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{35 e \left (e x\right )^{\frac{5}{2}}} - \frac{8 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- \frac{21 A c x}{2} + \frac{25 B a}{2}\right )}{105 e^{2} \left (e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(7/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.19484, size = 254, normalized size = 0.68 \[ \frac{x \left (16 a^{3/2} c x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (63 A \sqrt{c}+25 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-1008 a^{3/2} A c^{3/2} x^{7/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (7 a^2 (3 A+5 B x)-4 a c x^2 (63 A+20 B x)-3 c^2 x^4 (7 A+5 B x)\right )\right )}{105 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{7/2} \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(7/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.029, size = 367, normalized size = 1. \[{\frac{2}{105\,{x}^{2}{e}^{3}} \left ( 504\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c-252\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c+100\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}+15\,B{c}^{3}{x}^{7}+21\,A{c}^{3}{x}^{6}+95\,aB{c}^{2}{x}^{5}-231\,aA{c}^{2}{x}^{4}+45\,{a}^{2}Bc{x}^{3}-273\,{a}^{2}Ac{x}^{2}-35\,{a}^{3}Bx-21\,A{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(7/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(7/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(7/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(7/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(7/2),x, algorithm="giac")
[Out]