Optimal. Leaf size=190 \[ -\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{d^2 \sqrt{a+b x}}+\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d^2 \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.207322, antiderivative size = 190, 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, Rules used = {47, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d^2 \sqrt{a+b x}}+\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{d^2 \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{(c+d x)^{5/4}} \, dx &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}+\frac{(2 b) \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}+\frac{(8 b) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2}\\ &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}-\frac{\left (8 \sqrt{b} \sqrt{b c-a d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2}+\frac{\left (8 \sqrt{b} \sqrt{b c-a d}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2}\\ &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}-\frac{\left (8 \sqrt{b} \sqrt{b c-a d} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2 \sqrt{a+b x}}+\frac{\left (8 \sqrt{b} \sqrt{b c-a d} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2 \sqrt{a+b x}}\\ &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}-\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \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 )}{d^2 \sqrt{a+b x}}+\frac{\left (8 \sqrt{b} \sqrt{b c-a d} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d^2 \sqrt{a+b x}}\\ &=-\frac{4 \sqrt{a+b x}}{d \sqrt [4]{c+d x}}+\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \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 )}{d^2 \sqrt{a+b x}}-\frac{8 \sqrt [4]{b} (b c-a d)^{3/4} \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 )}{d^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.041427, size = 73, normalized size = 0.38 \[ \frac{2 (a+b x)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac{5}{4},\frac{3}{2};\frac{5}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \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{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{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{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, 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{\sqrt{a + b x}}{\left (c + d x\right )^{\frac{5}{4}}}\, 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{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]