3.3142 \(\int \frac{(a+b x)^{4/3}}{\sqrt{c+d x} (e+f x)} \, dx\)

Optimal. Leaf size=100 \[ \frac{3 (a+b x)^{7/3} \sqrt{\frac{b (c+d x)}{b c-a d}} F_1\left (\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{7 \sqrt{c+d x} (b e-a f)} \]

[Out]

(3*(a + b*x)^(7/3)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*AppellF1[7/3, 1/2, 1, 10/3, -
((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(7*(b*e - a*f)*Sqrt[
c + d*x])

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Rubi [A]  time = 0.275224, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 (a+b x)^{7/3} \sqrt{\frac{b (c+d x)}{b c-a d}} F_1\left (\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{7 \sqrt{c+d x} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(4/3)/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(3*(a + b*x)^(7/3)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*AppellF1[7/3, 1/2, 1, 10/3, -
((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(7*(b*e - a*f)*Sqrt[
c + d*x])

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Rubi in Sympy [A]  time = 23.2661, size = 85, normalized size = 0.85 \[ \frac{3 b \left (a + b x\right )^{\frac{7}{3}} \sqrt{c + d x} \operatorname{appellf_{1}}{\left (\frac{7}{3},\frac{1}{2},1,\frac{10}{3},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{7 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \left (a d - b c\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(4/3)/(f*x+e)/(d*x+c)**(1/2),x)

[Out]

3*b*(a + b*x)**(7/3)*sqrt(c + d*x)*appellf1(7/3, 1/2, 1, 10/3, d*(a + b*x)/(a*d
- b*c), f*(a + b*x)/(a*f - b*e))/(7*sqrt(b*(-c - d*x)/(a*d - b*c))*(a*d - b*c)*(
a*f - b*e))

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Mathematica [B]  time = 5.09939, size = 921, normalized size = 9.21 \[ \frac{6 b \sqrt{c+d x} \left (\frac{7 d (a+b x)}{f}+\frac{(c+d x) \left (-26 (b c-a d) (3 b d e+2 b c f-5 a d f) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right ) \left (b (3 c f-3 d e) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 (b c-a d) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )-7 b (c+d x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right ) \left (13 f \left (-\left (12 f c^2+2 d (16 e+7 f x) c+35 d^2 e x\right ) b^2+a d (-3 d e+42 c f+49 d f x) b+5 a^2 d^2 f\right ) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+14 (5 b d e+2 b c f-7 a d f) \left (3 b (d e-c f) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+2 (a d-b c) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )\right )\right )}{d (e+f x) \left (7 b f (c+d x) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (6 c f-6 d e) F_1\left (\frac{7}{6};\frac{2}{3},2;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+4 (b c-a d) f F_1\left (\frac{7}{6};\frac{5}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right ) \left (13 b f (c+d x) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+b (6 c f-6 d e) F_1\left (\frac{13}{6};\frac{2}{3},2;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+4 (b c-a d) f F_1\left (\frac{13}{6};\frac{5}{3},1;\frac{19}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )\right )}\right )}{35 d^2 (a+b x)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^(4/3)/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(6*b*Sqrt[c + d*x]*((7*d*(a + b*x))/f + ((c + d*x)*(-26*(b*c - a*d)*(3*b*d*e + 2
*b*c*f - 5*a*d*f)*AppellF1[7/6, 2/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e)
+ c*f)/(f*(c + d*x))]*(b*(-3*d*e + 3*c*f)*AppellF1[7/6, 2/3, 2, 13/6, (b*c - a*
d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 2*(b*c - a*d)*f*AppellF1[7/6,
5/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]) - 7*b*(c
+ d*x)*AppellF1[1/6, 2/3, 1, 7/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*
(c + d*x))]*(13*f*(5*a^2*d^2*f + a*b*d*(-3*d*e + 42*c*f + 49*d*f*x) - b^2*(12*c^
2*f + 35*d^2*e*x + 2*c*d*(16*e + 7*f*x)))*AppellF1[7/6, 2/3, 1, 13/6, (b*c - a*d
)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 14*(5*b*d*e + 2*b*c*f - 7*a*d*f
)*(3*b*(d*e - c*f)*AppellF1[13/6, 2/3, 2, 19/6, (b*c - a*d)/(b*c + b*d*x), (-(d*
e) + c*f)/(f*(c + d*x))] + 2*(-(b*c) + a*d)*f*AppellF1[13/6, 5/3, 1, 19/6, (b*c
- a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]))))/(d*(e + f*x)*(7*b*f*(c +
d*x)*AppellF1[1/6, 2/3, 1, 7/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c
+ d*x))] + b*(-6*d*e + 6*c*f)*AppellF1[7/6, 2/3, 2, 13/6, (b*c - a*d)/(b*c + b*
d*x), (-(d*e) + c*f)/(f*(c + d*x))] + 4*(b*c - a*d)*f*AppellF1[7/6, 5/3, 1, 13/6
, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))])*(13*b*f*(c + d*x)*Ap
pellF1[7/6, 2/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x)
)] + b*(-6*d*e + 6*c*f)*AppellF1[13/6, 2/3, 2, 19/6, (b*c - a*d)/(b*c + b*d*x),
(-(d*e) + c*f)/(f*(c + d*x))] + 4*(b*c - a*d)*f*AppellF1[13/6, 5/3, 1, 19/6, (b*
c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))]))))/(35*d^2*(a + b*x)^(2/3
))

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int{\frac{1}{fx+e} \left ( bx+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(4/3)/(f*x+e)/(d*x+c)^(1/2),x)

[Out]

int((b*x+a)^(4/3)/(f*x+e)/(d*x+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{\sqrt{d x + c}{\left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(4/3)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(4/3)/(sqrt(d*x + c)*(f*x + e)), x)

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Fricas [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)^(4/3)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\sqrt{c + d x} \left (e + f x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(4/3)/(f*x+e)/(d*x+c)**(1/2),x)

[Out]

Integral((a + b*x)**(4/3)/(sqrt(c + d*x)*(e + f*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{\sqrt{d x + c}{\left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(4/3)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="giac")

[Out]

integrate((b*x + a)^(4/3)/(sqrt(d*x + c)*(f*x + e)), x)