3.1640 \(\int \frac{(c+d x)^{3/4}}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=221 \[ -\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{d (c+d x)^{3/4}}{b \sqrt{a+b x} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}} \]

[Out]

(-2*(c + d*x)^(3/4))/(3*b*(a + b*x)^(3/2)) - (d*(c + d*x)^(3/4))/(b*(b*c - a*d)*
Sqrt[a + b*x]) + (d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4)
*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(7/4)*(b*c - a*d)^(1/4)*Sqrt[a + b
*x]) - (d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)
^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(7/4)*(b*c - a*d)^(1/4)*Sqrt[a + b*x])

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Rubi [A]  time = 0.675101, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{d \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{7/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{d (c+d x)^{3/4}}{b \sqrt{a+b x} (b c-a d)}-\frac{2 (c+d x)^{3/4}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/4)/(a + b*x)^(5/2),x]

[Out]

(-2*(c + d*x)^(3/4))/(3*b*(a + b*x)^(3/2)) - (d*(c + d*x)^(3/4))/(b*(b*c - a*d)*
Sqrt[a + b*x]) + (d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticE[ArcSin[(b^(1/4)
*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(7/4)*(b*c - a*d)^(1/4)*Sqrt[a + b
*x]) - (d*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)
^(1/4))/(b*c - a*d)^(1/4)], -1])/(b^(7/4)*(b*c - a*d)^(1/4)*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 80.8208, size = 406, normalized size = 1.84 \[ \frac{d \left (c + d x\right )^{\frac{3}{4}}}{b \sqrt{a + b x} \left (a d - b c\right )} - \frac{2 \left (c + d x\right )^{\frac{3}{4}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{d^{2} \sqrt [4]{c + d x} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{7}{4}} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{d \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{7}{4}} \sqrt [4]{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/4)/(b*x+a)**(5/2),x)

[Out]

d*(c + d*x)**(3/4)/(b*sqrt(a + b*x)*(a*d - b*c)) - 2*(c + d*x)**(3/4)/(3*b*(a +
b*x)**(3/2)) - d**2*(c + d*x)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d)/(b**(3/2)*(
a*d - b*c)**(3/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)) + d*sqrt((a*d - b
*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)**2))*
(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_e(2*atan(b**(1/4)*(c + d*x)
**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(b**(7/4)*(a*d - b*c)**(1/4)*sqrt(a - b*c/d +
b*(c + d*x)/d)) - d*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c
+ d*x)/sqrt(a*d - b*c) + 1)**2))*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*ell
iptic_f(2*atan(b**(1/4)*(c + d*x)**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(2*b**(7/4)*(
a*d - b*c)**(1/4)*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.227657, size = 104, normalized size = 0.47 \[ \frac{(c+d x)^{3/4} \left (-d (a+b x) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+a d+2 b c+3 b d x\right )}{3 b (a+b x)^{3/2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/4)/(a + b*x)^(5/2),x]

[Out]

((c + d*x)^(3/4)*(2*b*c + a*d + 3*b*d*x - d*(a + b*x)*Sqrt[(d*(a + b*x))/(-(b*c)
+ a*d)]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(3*b*(-(b
*c) + a*d)*(a + b*x)^(3/2))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{3}{4}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/4)/(b*x+a)^(5/2),x)

[Out]

int((d*x+c)^(3/4)/(b*x+a)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(3/4)/(b*x + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(5/2),x, algorithm="fricas")

[Out]

integral((d*x + c)^(3/4)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x\right )^{\frac{3}{4}}}{\left (a + b x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(3/4)/(b*x+a)**(5/2),x)

[Out]

Integral((c + d*x)**(3/4)/(a + b*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{3}{4}}}{{\left (b x + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/4)/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(3/4)/(b*x + a)^(5/2), x)