3.1638 \(\int \frac{(c+d x)^{3/4}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=196 \[ -\frac{12 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt{a+b x}}+\frac{12 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4}}{5 b} \]

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*b) + (12*(b*c - a*d)^(7/4)*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])/(5*b^(7/4)*d*Sqrt[a + b*x]) - (12*(b*c - a*d)^(7/4)*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]
)/(5*b^(7/4)*d*Sqrt[a + b*x])

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Rubi [A]  time = 0.652876, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{12 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt{a+b x}}+\frac{12 (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 b^{7/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*x]*(c + d*x)^(3/4))/(5*b) + (12*(b*c - a*d)^(7/4)*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])/(5*b^(7/4)*d*Sqrt[a + b*x]) - (12*(b*c - a*d)^(7/4)*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]
)/(5*b^(7/4)*d*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 65.8739, size = 384, normalized size = 1.96 \[ \frac{4 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{5 b} - \frac{12 \sqrt [4]{c + d x} \sqrt{a d - b c} \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}}{5 b^{\frac{3}{2}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )} + \frac{12 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{7}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} - \frac{6 \sqrt{\frac{a d - b c + b \left (c + d x\right )}{\left (a d - b c\right ) \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right )^{2}}} \left (a d - b c\right )^{\frac{7}{4}} \left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{a d - b c}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{7}{4}} d \sqrt{a - \frac{b c}{d} + \frac{b \left (c + d x\right )}{d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(a + b*x)*(c + d*x)**(3/4)/(5*b) - 12*(c + d*x)**(1/4)*sqrt(a*d - b*c)*sqr
t(a - b*c/d + b*(c + d*x)/d)/(5*b**(3/2)*(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c)
+ 1)) + 12*sqrt((a*d - b*c + b*(c + d*x))/((a*d - b*c)*(sqrt(b)*sqrt(c + d*x)/sq
rt(a*d - b*c) + 1)**2))*(a*d - b*c)**(7/4)*(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)/(5*
b**(7/4)*d*sqrt(a - b*c/d + b*(c + d*x)/d)) - 6*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))*(a*d - b*c)**(7/4)*
(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c) + 1)*elliptic_f(2*atan(b**(1/4)*(c + d*x)
**(1/4)/(a*d - b*c)**(1/4)), 1/2)/(5*b**(7/4)*d*sqrt(a - b*c/d + b*(c + d*x)/d))

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Mathematica [C]  time = 0.159962, size = 93, normalized size = 0.47 \[ \frac{4 (c+d x)^{3/4} \left ((b c-a d) \sqrt{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{5 b d \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*(c + d*x)^(3/4)*(d*(a + b*x) + (b*c - a*d)*Sqrt[(d*(a + b*x))/(-(b*c) + a*d)]
*Hypergeometric2F1[1/2, 3/4, 7/4, (b*(c + d*x))/(b*c - a*d)]))/(5*b*d*Sqrt[a + b
*x])

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{3}{4}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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